The question is wider ranging than might first be thought...
Free power compared to what? The same engine that doesn't have a turbo? At full throttle, part throttle, a certain rpm range? Can a smaller engine with a turbo make the same power as a bigger engine without a turbo while using less fuel than the big engine?
My general thoughts go..
There are a lot of if/buts and dynamics at play...
For starters a turbo'd engine will usually run lower compression than a N/A engine, they have to to run lower compression to prevent detonation of the charge when running boost. The lower compression makes the engine less efficient in most conditions when the engine isn't under boosted manifold pressure. Actually that's not quite the right way to think about it because lower compression actually aids fuel economy because compressing air in engine cylinders uses power and the more we compress it the more we loose to compression... but with increased compression comes increased expansion ratio and to get the best efficiency from the combustion/expanding gas in the cylinders a higher expansion ratio is better. We want a high expansion ratio and in order to get it we need a high compression ratio because the 2 are linked, they're not identical in an engine application because an engine has valve timing and intake flow / exhaust flow dynamics that can make them different (especially on a modern engine with VVT) but they are still inextricably linked in an engine application.
If you mean can bolting a turbo onto an engine make it more powerful without doing anything else (other than changing fuelling and ignition maps) then yes it can. But still you have to pay for the turbo. And there's still a caveat.. If we can bolt a turbo onto an engine and not do anything else to it for that engine to make more power it begs the question why didn't we need to lower compression (expansion) so the engine didn't suffer detonation when running under boost? We may have been able to get a bit more horsepower at full engine load, plus an increase in fuel economy across the entire engine load range, just by raising the compression (expansion) ratio. Raising the compression (expansion) ratio doesn't add much power (nothing like adding a turbo) or make a dramatic difference to fuel economy but it's something that's probably more likely to make truly 'free power' than turbos all other things being the same (gearing etc), at least at light engine loads. The point here is that if we modified an engine setup by fitting a turbo without doing anything else and found that we could get X amount more power from it by doing so, we'd have to compare the new 'free' engine power to the free power we could have got if we hadn't fitted a turbo but had increased it's compression. Almost for sure 'if we didn't do anything else' (such as change the gearing) at part loads the high compression modification engine could do better mpg than the turbo modification engine - perhaps fuel economy at part throttle loads is another take on 'free power'.
Back pressure in an exhaust system is a negative for power, it amounts to both a pumping loss (pistons in cylinders on exhaust strokes are having to push against that pressure) and inlet charge dilution with exhaust gas.. any pressure left in cylinders when the exhaust valve is closing and inlet valve opening is pressure that works against new inlet charge entering the cylinder when the piston starts it's induction stroke. Whenever there's backpressure in an exhaust system it's the same effect as the engine having an EGR valve open constantly, instead of the cylinder getting a completely fresh charge it gets a charge that is part exhaust gas and part fresh inlet charge. Think about it this way.. Turbo's use back pressure to spin the exhaust turbine, which spins the compressor turbine. There's more exhaust gas going through the exhaust turbine (and it's hotter) than there is going through the compressor turbine (which is colder), so it might be possible for a very efficient turbo to make more boost pressure in the compressor than it causes as back pressure in the exhaust, but in practice they don't. In practice a turbo causes more backpressure in the exhaust than boost pressure it creates. In practice despite the greater volume and higher energy (temperature) of the exhaust flow a turbo will still cause more backpressure in the exhaust than boost pressure it creates. This means that when the exhaust valve is closing and inlet valve opening there's a period when no new charge can enter the cylinder because the cylinder still contains a higher pressure than the inlet charge and this period extends until the piston is quite a way down on it's induction stroke, in fact during this period exhaust flows into the inlet manifold!
Some things that work in a turbo's favour are...
If we bolt a turbo onto an engine we could use taller gearing, so during (say) cruise conditions we could run the engine at lower rpm, less frictional losses, less vacuum pumping losses on induction strokes, less fuel used. We might not want to run the taller gearing without the turbo because during our cruise a slight uphill stretch could necessitate a downchange in gear, the taller gearing could spoil the driveability of the car. With the turbo we can run the taller gearing because when we get to the hill we put our foot down a little, we get turbo boost and enough torque to go uphill without the downchange in gear.
If we were running say a 5L n/a engine we could maybe fit a 2.5 turbo engine which could make the same power whilst using less fuel at part engine loads even if it does run lower compression than the V8.
A coincidence you asked about turbos because only last night I watched a David Vizard video on YouTube on the subject. He confirmed what I already knew about backpressure playing against fresh inlet charge at the end of the exhaust stroke but he went a bit further and actually put some numbers to backpressure versus boost pressure. He was approaching the subject more from a cams (valve timing) point of view but a very interesting video. I've got some of his books but it was the first time I knew he made videos on Youtube (thought he was dead by now lol)
https://www.youtube.com/watch?v=506pjO4Z_80
I think exhaust scavenging (the idea of 'pulses' in the exhaust system 'pulling' more exhaust gas out of cylinders when exhaust valves open than would otherwise just be pushed out due to high pressure in cylinders escaping when an exhaust valve opens) has a smaller positive effect than backpressure has negative effect. For sure there are some engines that won't even run properly unless an exhaust system is fitted that creates some backpressure but I believe that's due to the cam timing they run with that features a lot of either/both the exhaust valve opening early (before BDC on the power stroke) or remaining open for longer than average during the intake stroke... They run such cam timing because the cylinder head design isn't great and to make as much power as they do they need to run such cam timing. The backpressure on them does things like preventing inlet charge flowing out of the exhaust... But they could be made to run better if only the comubustion chamber design / ports design were better and with more standard cam timing. When they run some engines on engine dynos they only fit a header (no exhaust system) and the power they produce isn't much different to when an exhaust system is fitted across the entire rpm / load range, unless it's an engine design that needs the backpressure to run properly (like say an old VW Beetle engine). I do get the thing about selecting an exhaust system diameter and length to tune the resonance of exhaust pulses so that the one in front is pulling the one behind out (I liken this to filling a bottle with water and emptying the bottle as quickly as possible by holding it at a specific angle that works best for getting air in and water out, kind of a very loose analogy!) but in this same analogy not much water would come out if it were a narrow tube with a big air bubble at the top inside the tube and air pressure outside the bottle was high enough to want to push the water up into the bubble, at least not until the bubble were compressed to be at near same pressure as the pressure outside the tube. With a good combustion chamber and port design I reckon back pressure is going to have much more detrimental effect than scavenging has positive effect, particularly on a turbo where there's a turbine separating the 'tuned' exhaust system from the pulses from exhaust valves. Turbos do indeed quieten an exhaust as much as an exhaust box does but in doing so they interfere with the 'sonic/pressure' pulses coming from exhaust ports on the engine. With the pulses gone 'pulse scavenging' is also gone, then you're just left with decreasing exhaust back pressure on the exhaust (as opposed to engine) side of the turbo exhaust turbine as a means to increase power - and at least if the intention is to make the most power with the minimum boost possible while being kind to the engine you want the turbo backpressure to be as low as possible in comparison to boost pressure.
There's still a lot more to it... Would probably be possible to build a very high powered turbo that was more economical at part load than a normal turbo, but the normal turbo would probably have better driveability with the torque coming in (and with less turbo lag) at lower rpms compared to the high powered more economical turbo. When building your turbo engine you'll have a power figure in mind, the power figure will affect/dictate what compressor housing your turbo has. Then you need an exhaust turbine that will power (turn) the compressor housing. Whether you fit a relatively small/medium/large exhaust housing, and it's efficiency, will affect the rpm/load that boost starts from and the backpressure to boost pressure ratio. Each aspect affects all others, so backpressure versus boost pressure will affect which cam you should choose, which cam will obviously affect everything else anyway including max boost and at what rpm max boost occurs. You could probably even select a setup to effect where during the combustion stroke most of the power is made, with a high backpressure to boost ratio that's going to be lower in the power stroke than it would be with a lower backpressure to boost ratio and that might effect things like engine life because high piston loads half way down a stroke impart greater sideways loading on the piston in it's cylinder than loads near TDC and BDC.
Says me while currently struggling to rebuild my boat's small block Ford 5.0 engine with it's old-skool pushrod design lol! But I know that if I fitted more modern design cylinder heads with a bit better combustion chamber design and better ports I could get another 100bhp with everything else staying the same (same cam and timing etc). Or if I rebuilt it with some cams while keeping the old cast heads it might need some backpressure to run OKish (not as smooth as it runs now) with the extra 100bhp. Or I could bolt turbo(s) on it but would probably have to lower the compression, which would make it less economical at part engine loads, though maybe still make more power than a similar design engine that didn't run turbo(s) while using less fuel than the bigger engine. As said at the begginning much of it depends on exactly what you are trying to achieve and 'free power' depends on what you're comparing it to. I could get a bit more power and economy just by increasing it's compression with the standard cast heads.
I agree with Gilbert on
1. It could spin for almost free once it has been 'spun up to speed'. It will take some power to spin up to speed.
2. It will add some sort of restriction. But we haven't defined the specifics of the fan.. This could be a tiny fan, hardly any restriction regardless of it's design (at the small size), could be full diameter of the exhaust but with extremely narrow blades. Fan isn't a bad analogy but a turbo has turbines not fans, slight difference.
3. Less power due to an exhaust restriction if in 1 or 2 we draw any power from the exhaust gas. But we're overlooking the real power draw due to the backpressure caused by the 'fan' in a turbo's exhaust, the power draw (and hence exhaust restriction) needed to power the turbo's compressor housing... If we put a load on the fan and slow it down it's going to be more of a restriction and cause more back pressure. But a turbo'd engine that has a turbo with an electronically controlled wastegate could be set up for exhaust gas to bypass the turbo during very light engine load / steady cruise conditions anyway... by keeping the turbo's wastegate open. Still the turbo'd engine would likely have lower compression (ahem expansion) than it's n/a equivalent... but it could be higher geared, could be a smaller engine.
I saw it coming many years ago - Manufacturers are making smaller engines than before, turbo'd so they make the same power as older n/a engines but with better fuel economy. A 1 litre Ford Mondeo Ecoboost or a 2.3L Ford mustang Ecoboost, as opposed to a 2L n/a Mondeo or a 5.0 V8 n/a Mustang. Will those newer engines last as long as the older engines? Probably not because the engines are going to be under higher loads (related to their size) than the older engines, they only make as much power as the older bigger engines so are going to have to work under boost conditions (higher loads than the bigger engines) more of the time just in normal driving. But can a 2L turbo engine or 5L turbo engine last as long as a 2L n/a or 5L n/a engine? Yes depending how it's driven.. It doesn't need to be driven any harder than the older design, just that it can be driven harder and make more power. My old Sierra Cosworth with 400bhp from it's Pinto derived engine and still capable of 30+ mpg definitely felt like it had free power (considering a 2L Sierra wouldn't do a great deal better than 30mpg anyway) at low engine loads on a cruise but mpg could go through the floor if I gave it some stick, in fact I might as well have fitted a tuned V8 if I wanted to boot it all the time, bet it would be just as economical during booting it. Every engine or drivetrain setup is a compromise but 30mpg from a 2L with 400bhp (Sierra Cosworth) seems a good compromise compared to 36mpg with 130bhp (Sierra 2L n/a). A 1L Ecoboost engine makes 123bhp standard, close to the 130bhp n/a 2L. If we fitted a 1L engine in the Sierra and drove it slowly we'd expect better than the 2L mpg but it would be a very sluggish car. But fit a turbo on it, make it 123bhp and it won't be far from the 2L power and responsiveness and even with the turbo it will still do better mpg than the 2L mpg while driving similarly to the 2L.
Fitting a turbo to an engine increases it's overall operating dynamic range. By this I mean the amount of fuel it consumes just to idle compared to the amount of fuel it consumes at maximum engine load (making it's max power). If we level out / normalise all the variables so we can clearly see this dynamic range, an n/a engine might idle at 700rpm and pulse it's injectors at 3ms at idle, so let's say the idle fuel useage is 700x3 = 2100 units at idle (no need to define a unit to this 2100 at the moment, even though there actually is one which we could/can define). At max bhp this engine might be doing 16ms injector duration and 6000rpm, which is 6000x16 = 96000 units. The dynamic range would be 96000/2100 = 45.7, i.e. it would use 45.7 times as much fuel flat out and making max bhp than it uses at idle. Now let's suppose this engine makes 200bhp. So if we do 96000/2100 we can work out that at idle it's using 4.375bhp worth of fuel. Now let's look at likely figures for a very similar design engine only with the expected slightly lower compression, obviously a turbo stuck in it's exhaust and making 300bhp from (say) 0.7 bar of boost (say something like a 2L version of a Saab LPT light pressure turbo engine). The turbo engine still makes max bhp (now 300bhp) at the same 6000 rpm, it still has the same 'injection window' (period of time during which the engine completes it's 4 stroke cycle) at 6000rpm as any 4 stroke engine at 6000rpm so it still can only go up to a peak of 16ms injector duration pulse time, so with a need to inject more fuel in the same time frame (so we don't overshoot the window) we need to increase the amount of fuel the inectors flow when they are open... If all else remains the same (efficiency etc) then we need the injectors to flow 1.5x the fuel they'd flow for 200bhp to flow enough for 300bhp. We could get the injectors to flow 1.5x the fuel by increasing the fuel pressure, but we'd have to more that double fuel pressure and that kind of fuel pressure wouldn't be safe, or we could fit bigger flowing injectors... so we fit bigger flowing injectors. Then, again all else being the same (efficiency at idle) these bigger flowing injectors would need to be pulsed for less duration than the lower flowing injectors to deliver the same amount of fuel. Where before with the 200bhp setup we needed to pulse injectors for 3ms at idle, now (if injectors opened and closed instantaneously, which they don't) we'd only have to pulse them for 2ms. Now if we have an engine that idles at 700 rpm with 2ms duration and reaches peak power (of 300bhp) at 6000rpm and 16ms durtation we have dynamic range of (700 x 2) = 1400 compared to (6000 x 16) = 96000, so dynamic range for the turbo is now 68.57 compared to 45.7 for the n/a... and if we compared 68.57 to 45.7 we'd find the same 1.5 ratio which is the same as the power increase (from 200 to 300bhp). But we haven't accounted for the lower efficiency at part loads due to the lower compression, and what we'd actually find is that the petrol injectors on the turbo engine would probably have duration at closer to 2.1 or 2.2ms at idle - which you should be thinking is due to the lower compression engine needing more fuel (because it needs to use more air and it has to keep the mixture correct or things look even worse) just to idle... In fact it needs more air and fuel to deliver the same power as the higher compression n/a engine to deliver any sort of low power level. But it will make 50% more power when you put your foot down, and we could compare that bit of extra fuel usage (10%>20%?) at low engine loads to the 50% more power when do put our foot down and consider that without the turbo we'd need a 50% bigger engine (go up in size from 2L to 3L with same spec) and the 3L engine might have higher fuel usage at low engine loads (almost certainly at idle) than the 2L turbo.