Fans like in the hair dryer are not quite like fans in a jet engine. In the hair dryer you can only narrow up the nozzle so much. After that the hair dryer fan cannot push air through like it did when the nozzle was open. We start to lose correct results at that point. Well we did demonstrate one point, we removed some energy from the hot air and converted it into a faster stream of air.
Turbines have the ability to also extract energy from the stream of gases in a jet engine.
They do this by converting the kinetic energy or enthalpy of the flowing fluid into mechanical work. Turbine operation is a a process and not a cycle. In fact the gas turbine engine operates a series of processes and not cycles. When the gases from the combustion chamber are directed to the turbine they first meet up with nozzles that speed them up as we now know. Actually in a jet engine at this point the gases have been sped up to near the speed of sound.