A388,
My take on this phenomenon you're asking about is that all piston engines lose power as density altitude increases - it's the nature of the beast.
Turbo charging as well as increasing power for T/O also maintains a higher air pressure in the induction system, ie a lower density altitude so you have more power. Hence better take off performance on hot, high D/A, days. Eventually you "run out of puff" as altitude increases and the turbo is maxed out so D/A starts to increase with further climb. A benefit of turbos is the recovery of waste energy going down the exhaust pipe, especially heat which is recovered as the exhaust gasses expand through the turbine. But nothing for nothing and it costs due to higher back pressure in the exhaust reducing overall efficiency a bit. Superchargers (mechanically driven) have similar benefits excluding the heat recovery but they tend to cost a bit more energy to run.
Turbos can and often are optimised for the mission so sizes vary on otherwise similar engines - look at engines powering, say, a Beech Duke and a Piper Chieftain, different planned cruise regimes mean different turbo sizing. Look also at Piper turbo normalised Seminoles which use small turbos mainly to help take off performance as D/A rises.