I think the problem stems from understanding what we mean by ‘compressing’ the air, basically a turbocharger or supercharger is just a fan that vacuums up atmospheric air and presses it into the intake manifold. The basics are all the same, as the cylinder goes down and creates a low pressure area (because you had one area and then you retracted the piston creating a greater area with the same amount of total air) high pressure air wants to go fill that void. If you have a supercharger/turbocharger, that high pressure air is simply higher pressure than it would be without one because you vacuumed a bunch of air molecules into the intake manifold.
It plays the same essential role as the low pressure compression stages in a gas turbine like a turbofan engine; the air needs to be moving nice and fast and with a lot of force to turn the turbine, so you get a bunch of bladed fans together to compress the air using the energy of the escaping gases to turn a turbine that is connected to the same shaft as the compression stages and ultimately the big fan at the front of the turbine.
The difference with turbochargers and superchargers is that instead of being connected to a shaft like in a turbine engine, it is connected to the exhaust or crankshaft which delivers the energy required to spin the fans to create that pressurization.
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