From windmill physics we see that wind power is proportional to the cube of windspeed as:
Wind power = Density × Area × v$^3$/2
The exhaust gasses of a jet engine are of lower density from heating, slightly higher mass, but are going much faster than the inlet gasses
This explains why a smaller turbine can run a larger (more blade area compresser) with plenty of energy left over for propulsion.
How to use excess energy?
The most efficient method of propulsion depends on the desired airspeed.
At low airspeeds propellers (literally rotating wings) are most efficient at generating thrust. Energy is extracted to turn them by adding more turbine blade area. This reduces exhaust gas velocity but extracts more mechanical energy.
At higher airspeeds the fan replaces the propeller. At transsonic and supersonic speeds, the "extra windmill" can be removed from the turbine and the engine operates as a "turbojet", essentially an airbreathing rocket with the turbine only running the compresser.
Faster still, one can remove the compresser and turbine and operate as a ramjet.