The answer as almost always is, it depends. As most aviation text books or manuals will tell you a turbo charger uses exhaust gas to drive a turbine which drives a compressor which compresses air into the cylinder for combustion. Basic combustion principles will tell us that there is an optimum fuel to air mixture required for efficient fuel burn in the cylinder. So naturally when we increase the amount of air mixture by compressing more pieces of air into a cylinder, we also increase the amount of fuel we need to make the whole process work... (efficiently as possible). So its easy to see why we can increase power output with a turbo charger, but where do the efficiencies come into play?
That is a good question, lets look at an example. Lets compare a 160 HP naturally aspirated engine with its turbo charged counterpart. Down at sea level a naturally aspirated engines cylinder at full throttle will have close to 29.92 inches of mercury of pressure pushing air molecules into our cylinder, where we then apply the appropriate fuel/air mixture for good combustion. At this point our engine is producing all 160 horses of power.
Compare this with the equivalent engine but turbo charged and rated for the same power. At sea level, can it use its turbo charger to produce anymore power? The awnser is yes and no. "Yes" it could but it might destroy the engine, so in reality "no" its still limited to 160 hp, so in this case the turbo can't push anymore air into the cylinder than is already being driven into there by the ambient air pressure at sea level. So at sea level there is little benefit to a turbo charger.
Now if we take our comparison up to 10,000 feet where the standard pressure is 19.92 inches our naturally aspirated engine can only produce 19.92 inches of manifold pressure of power. That's a lot less, however keep in mind it is burning a proportionally smaller amount of fuel. For that smaller fuel burn it is also getting a slower indicated airspeed (the air resistance it feels), than it might down low at higher power settings. However, its getting a higher “true” airspeed (you have to go faster through thin air to get the same “indicated” air resistance on the pitot tube). So we accept the slower indicated airspeed at higher altitude in trade for a higher true airspeed. This explains why we go higher for better efficiencie no matter the engine. Unfortunately with our naturally aspirated engine there is a limit to how high we can go to take advantage of this greater and greater differential between indicated airspeed and true airspeed. Eventually we won't produce enough power in our oxygen deprived engine to keep us above stall speed (absolute altitude). In addition right above stall speed is not a very efficient speed with which to fly aerodynamically as well, further limiting the optimum altitude of our non turbo charged aircraft.
But what if we could somehow get that 160 hp to operate at 160 hp at higher altitudes, we could take advantage of the indicated and true airspeed split, and cover a lot more ground for the same cost. There in lies part of the answer to whether a turbo charger is worth it. If the aircraft is operated at low altitude, then chances are a naturally aspirated aircraft would be as fast and burn about the same amount of fuel than its turbo charged counterpart. Only difference would be the decreased cost of having to purchase and maintain the turbo. Now if your trip mission calls for the potential to get to higher altitudes 10000 feet and above you could gain large efficiency gains from a turbo.
Thats not all though in the question of wether a turbo is worth it. For example, take our 160 hp engine, previously we said it was only rated for 160 horses, but what if we could jam more air into the cylinder than would be possible in a non turbo charged airplane at sea level. What if we could increase the manifold pressure from the natural 29.92 sea level pressure up to a boosted 31 inches.
With a non turbo charged airplane this would be impossible unless we were to fly below sea level (not ideal), as the naturally aspirated engine depends upon outside pressure to push air into the cylinder for us to burn. So what would happen if we boosted it to 31 inches- awnser we would produce more power for a given size of engine. With this increased power the aircraft could possibly be rated for heavier gross weights and increased speed at higher altitudes. Further increasing not only our MPG at high altitudes but also our ability to carry more payload for the given aircraft further increasing efficiencies. Granted, we would burn slightly more fuel in the process but it would be made up for easily in our increased performance. Wether or not you could use the additional performance is another question entirely.
So ultimatly where is this increased efficiency coming from. Its tied to hot air. Literally. A turbo charger allows the operater to use this wasted heat coming from the exhaust to boost the amount of power it can produce from a given engine, thus increasing efficiency. However, at low altitudes this increase in efficiency is minimal compared to the efficiency obtained from a normal engine. It is at high altitudes this increased performance and efficiency is truly seen.