A turbocharger doesn't create energy. It's function is to increase the volumetric efficiency of the IC engine. How? By allowing more fuel to be pumped into the same size engine. It really is that simple.
Ideal combustion of gasoline occurs at the stoichiometric ratio, which is 14.7 to 1 (roughly, lots of variables). Let's call it 15 to 1.
That means you need 15 grams of air [1] for every gram of gasoline you pump into the engine.
How do you generate more power?
You burn more fuel per unit time.
How do you burn more fuel per unit time?
You build a bigger engine or you figure out how to pump more fuel into the same engine.
What is the limiting factor in being able to pump more fuel into the same engine?
You can't get enough air into a naturally aspirated engine in order to maintain the roughly 15 to 1 ratio. If you can't get more air into the engine you can't pump more fuel into it. Period.
How do we get more air into the engine?
You build a compressor. The compressor can be powered directly from the engine's drive shaft, in which case it is called a "supercharger", or it can be powered by exhaust gasses, which is known as a "turbocharger".
A turbocharger doesn't turn a piston engine into a turbine. It's function is to forcefully shove more air into the combustion chamber in order to allow a proportionally greater amount of fuel to be burned per unit time, thereby producing more power per engine unit volume, often termed "volumetric efficiency".
A greater volumetric efficiency does NOT mean an engine is more efficient. This is often a mistake made by consumers when buying cars. A turbocharger allows you to make more power with the same engine. It does not make it more efficient other than recovering some of the energy lost through exhaust.
Having owned several turbocharged and twin turbocharged fire-breathing monsters over the years I can tell you fuel efficiency was never better than non-turbo cars and, in my case at least, never a criteria for buying them. The latter said with a big fucking grin. And I have the speeding ticket collection to prove it.
officer: Do you know how fast you were going? I could not
catch up to you until I turned on the lights and
you slowed down.
driver (not to be identified):
Not sure. I stopped looking at the speedometer
at 120.
officer: If you want to get out of this ticket you have
to let me take this thing for a spin.
driver (now absolutely perplexed):
Here are the keys, sir
That was the longest uncomfortable 45 minute wait standing
by a CHP patrol car in the middle of the California desert.
The grin in the officer's face told the driver (not to be
identified) all would be well. Warning issued. Set cruise
control to 100 for the rest of the desert crossing.