there's more to it than that, GS.
The compressors are there to compress air, these are some of the turbine blades. They work just like the engine in your car...
Suck Squeeze Boom Blow
Air enters engine inlet...suck
Compressors mush the air into combustion chamber...squeeze
Ignition takes place in combustion chamber...boom
Burning fuel/air mixture expands and goes out the back of engine...blow
Turboprops don't leech energy from the turbines to turn the prop, they're connected via a gearbox. So the turbine speed is controlled by the throttles, the prop speeds are controlled by prop levers and there are also condition levers that regulate fuel flow into the turbine.
The turbine engines on turboprops aren't really big enough to power anything, they're essentially little APU motors.
Here is a short halfassed paper done on the Garrett engines for a B100 turboprop that I had to do to pass my 'Turbine Transition" ground course. It's a short overview done well enough to only pass but it may help.
AVSC 4200
King Air B100 Powerplants
The B100 uses Garrett TPE-331-6 Engines (everyone knows the –10’s an a 441 Conquest just ooze testosterone, but since I left my Conquest in my other pants I will stay on subject), which produce 715 Shaft horsepower each. There are five primary fuel injector nozzles, along with ten secondary nozzles for startup. 100% N1 in the B100 is equal to 41,730 turbine rpm and 2000 prop rpm…that in anyone’s book is a whole lot of spinning. The gearbox on the engine drives the Starter/Generator, fuel pumps, flow control unit, oil & scavenge pumps, tach generator and the propeller governor.
A power and engine speed lever located between the two pilots controls the engines. The power lever basically regulates how much fuel is going to the engine in order to produce the desired amount of power. The engine speed lever sets the governors. When the governors are engaged, they limit the power to a 65% minimum to a 97% maximum. When they are in “propeller governing mode” the prop governor controls the engine speed. This will keep engine speed within the 95-100% range.
The fuel cutoff and feather levers do just that. When selected, they shut off the fuel flow to the engine and feather the prop in case of an emergency. Friction locks are standard and enable the levers to be secured in any position desired.
Engine instruments on the panel include the ITT, Torque, Tach, Fuel Flow, and Oil temp & pressure gauges. These instruments are powered by 28 volts of DC power with the exception of the Tach (N1 Gauge), which is self-generating.