Why do turbine engines work?
The key is the combustion of fuel in the combustor. This adds energy to the flow so there is plenty available for the turbine to drive the compressor.
Depending on flight speed, the intake does already a fair amount of compression by decelerating the flow to Mach 0.4 - 0.5. More would mean supersonic speeds at the compressor blades, and the intake ensures a steady supply of air at just the right speed.
This speed, however, is far too high for ignition. The fuel needs some time to mix with the compressed air, and if flow speed is high, your combustion chamber becomes very long and the engine becomes heavier than necessary. Therefore, the cross section leading from the compressor to the combustion chamber is carefully widened to slow down the airflow without separation (see the section in the diagram below named "diffusor"). Around the fuel injectors you will find the lowest gas speed in the whole engine. Now the combustion heats the gas up, and makes it expand. The highest pressure in the whole engine is right at the last compressor stage - from there on pressure only drops the farther you progress. This ensures that no backflow into the compressor is possible. However, when the compressor stalls (this is quite like a wing stalling - the compressor vanes are little wings and have the same limitations), it cannot maintain the high pressure and you get reverse flow. This is called a surge.
The graph below shows typical values of flow speed, temperature and pressure in a jet engine. Getting these right is the task of the engine designer.
Plot of engine flow parameters over the length of a turbojet (picture taken from this publication)
The rear part of the engine must block the flow of the expanding gas less than the forward part to make sure it continues to flow in the right direction. By keeping the cross section of the combustor constant, the engine designer ensures that the expanding gas will accelerate, converting thermal energy to kinetic energy, without losing its pressure (the small pressure drop in the combustor is caused by friction). Now the accelerated flow hits the turbine, and the pressure of the gas drops in each of its stages, which again makes sure that no backflow occurs. The turbine has to take as much energy from the flow as is needed to run the compressor and the engine accessories (mostly pumps and generators) without blocking the flow too much. Without the heating, the speed of the gas would drop to zero in the turbine, but the heated and accelerated gas has plenty of energy to run the turbine and exit it at close to ambient pressure, but with much more speed than the flight speed, so a net thrust is generated.
The remaining pressure is again converted to speed in the nozzle. Now the gas is still much hotter than ambient air, and even though the flow at the end of the nozzle is subsonic in modern airliner engines, the actual flow speed is much higher than the flight speed. The speed difference between flight speed and the exit speed of the gas in the nozzle is what produces thrust.
Fighter engines usually have supersonic flow at the end of the nozzle, which requires careful shaping and adjustment of the nozzle contour. Read all about it here.
I just had an epiphany. The engine works because the turbine is "larger" than the compressor.
For extreme simplicity, let's assume that the working fluid is incompressible and effectively massless (it has pressure, but its inertia is negligible compared to the pressure). Assume further that the actual combustion is so finely tuned that the pressure stays constant during the combustion -- the gas simply expands at constant pressure, doing work against its own pressure as it does so.
Then the compressor and turbine really do operate across the same pressure differential, namely the difference between ambient pressure and pressure inside the combustion chamber.
At both ends of the engine, the power delivered to (or taken from) the drive shaft is the (common) pressure difference times the volume flow through the compressor/turbine. At this ideal level they are both the same kind of thing, except that one of them runs in reverse.
However, the torque is not necessarily the same. The turbine is constructed such that one revolution of the drive shaft will allow a certain volume of air to escape from the combustion chamber. (I suppose that is a matter of the turbine blades being mounted at a different angle than the compressor blades). At the other end of the shaft, one revolution of the shaft will push a certain smaller volume of air into the combustion chamber. It must be so because the gas expands during combustion.
This difference in volume-per-revolution means that the same pressure difference translates to different torques at the two ends of the engine.
As a completely idealized toy example we can imagine that the compressor and turbine are both made of the same kind of ideal reversible fan assemblies -- for each such unit, one crank of the handle will make a certain volume of air change places, and how hard the handle is to crank depends on the pressure difference.
The units that make up the compressor are mounted such that turning the drive shaft clockwise corresponds to air moving into the engine; the ones that make up the turbine are mounted opposite. Since the pressure difference is the same everywhere, the torque output from one turbine unit can drive exactly one compressor unit. But there are more turbine units than compressor units, and the additional ones produce surplus torque that can do work.
This corresponds to the fact that there's a net outflow of air from the combustion chamber, because new volumes of gas come into being as the fuel burns.
The air entering the combustion chamber from the compressor is moving at up to 600 mph. So when the fuel-air mixture burns and expands it has a choice of going upstream against a 600 mph wind or downstream through the turbine where there is relatively little resistance. Obviously it does the latter.
Jet engines are designed so the combustion doesn't raise the pressure in the combustion chamber very much. The exhaust gas flow out through the turbine is fast enough that the pressure in the combustion chamber remains low. Far too low to push the exhaust gases upstream and out through the compressor.