The advantage of the turbocharger is that its speed is determined by the density of the air it is compressing, to some extent independently of the engine rpm. This means as you fly higher, and the incoming air is less dense, it will rotate faster until the air exiting the compressor is at approximately the pressure it would be at a lower altitude.
The crankshaft driven supercharger on the other hand will run at the same speed regardless of altitude (for a given engine rpm) so its effectiveness reduces with altitude. The most efficient way round this would be a variable speed drive to the supercharger, so it speeds up with altitude. Some Merlins had a two speed supercharger as a way of partially overcoming this. An less efficient method is to dump some of the output of the compressor at low altitude, and close the dump valve as altitude increases.