Turbochargers allow an engine to burn more fuel and air by packing more into the existing cylinders in exactly the same way as a Supercharger. The difference being in the source of the power to drive the compressor. Whereas a Supercharger uses a belt or chain directly driven from the engine, a turbocharger uses a turbine driven by the pressurised exhaust gasses. The typical boost provided by a turbocharger is 6 to 8 psi. Since normal atmospheric pressure is 14.7 psi at sea level, you can see that you are getting about 50 percent more air into the engine. Therefore, you would expect to get 50 percent more power. It's not perfectly efficient, so you might get a 30- to 40-percent improvement instead.
One cause of the inefficiency comes from the fact that the power to spin the turbine is not free. Having a turbine in the exhaust flow increases the restriction in the exhaust. This means that on the exhaust stroke, the engine has to push against a higher back-pressure. This subtracts a little bit of power from the cylinders that are firing at the same time.
Turbochargers tend to be more efficient than Superchargers but usually more complicated to install, due to the special exhaust manifolds required and the extra heat considerations.