The F7F and F8F had single stage supercharged engines. Which means they had one stage of compression.
The engine on the P-47 had a similar supercharger connected to the engine, but with different gearing and only 1 ratio (F7F and F8F-1 had 2 ratios + neutral, F8F-2 had variable speed drive). The P-47 also had a turbocharger, which used exhaust gases to drive a compressor. So the P-47 had two stages of compression - the engine stage and the turbocharger.
As the air gets thinner at high altitudes, higher compression ratios in the supercharger are required. There is a point of compression ratios where the efficiency of a single stage falls away, which means eve more power is required to give the desired air flow and pressure to the engine. Adding a second stage means that each stage needs a lower compression ratio to achieve the desired overall compression ratio.
(Compression ratio = output air pressure/input air pressure)
In the P-51, from the B onward, this was achieved with a two stage supercharger driven by the engine. The 2 stage Merlin had the compressors on the same shaft, spinning at the same speed. The first compressor fed the second compressor, which fed the engine.
The engine has to drive the compressors, which costs power - up to a few hundred horsepower. The bigger the compression ratio required, the more power to drive the supercharger.
In the P-47 the turbocharger is that second compressor. The exhaust gases drive the turbine which drives the compressor. This doesn't cost much, if any, power at the crankshaft.
The turbocharger automatically compensates for altitude. It will allow the engine to maintain constant power up until the the critical altitude of the engine/turbocharger combination. One of the limiting factors there is the turbocharger's rpm.
The downsides of the turbocharger include the extra weight, extra space required, loss of exhaust thrust and complexity. For much of the war controlling the turbocharger was a problem.