The first thing you need to understand about combustion engines is that it's impossible for any ideal engine (even your car engine) to produce zero gross thrust. At the end of the thermodynamic cycle, the stagnation pressure and stagnation temperature of the working fluid are always greater than ambient, and so the flow always expands to a velocity greater than the free stream. This thrust is negligible in non-propulsion applications because of the exhaust design (and in fact net thrust can be negative when frictional effects are included), but the specifics of that topic are not germane to this discussion.
Secondly, we need to be more precise in our language. In particular, we need to highlight the very important distinction between static and stagnation fluid properties (static properties are independent of reference frame, while stagnation properties are not). This is very important because it is only static pressure gradients that cause changes in the local flow velocity. By design, the static pressure rises in every bladerow of the compressor, but the stagnation pressure increases only in the rotating bladerows. The static pressure falls in every bladerow of the turbine, but the stagnation pressure falls only in the rotating bladerows. You are correct in suggesting that the flow is pushing against the action of the compressor. This is because the flow is continually fighting an adverse pressure gradient; the flow is being pushed uphill against its natural inclination towards lower pressures. The turbine, on the other hand, is merely an obstruction as far as the flow is concerned, preventing it from expanding immediately down to the lowest pressure available (ambient). The flow doesn't mind doing some work for us, because it is being allowed to expand to lower static and stagnation pressures. Thus, a turbine can't really "stall" and reverse the flow direction in the way a compressor can. We can get away with the pressure rise in the compressor only because it occurs rather gradually (notice how little flow turning takes place in a compressor bladerow (<20 deg.) compared to a turbine (>100 deg.)). The flow does not shoot out of the front of the engine (normally) because the blades are meticulously designed to achieve the maximum compression possible before that happens (without universal success mind you, just ask any F-14 pilot before the GE-F110 engine upgrade).
Most relevant to your question, the stagnation pressure drop across the turbine is absolutely not the same as the pressure rise across the compressor it is mated to; it is always less. Why? Because for a given pressure ratio, the change in the stagnation enthalpy of a fluid increases with its initial temperature. Or, alternatively, the required pressure drop for a given chnage in enthalpy decreases as the initial temperature increases. The turbine powers the compressor, so the power produced by the turbine is consumed by the compressor (plus accessory drives). However, the turbine inlet temp is substantially greater than the compressor inlet temp, so the turbine exit stagnation pressure will necessarily be greater than the compressor inlet pressure, which is essentially equal to ambient.
If you are interested in the mathematical equations, the specific work (input) for the compressor and the specific work (output) of the turbine [kJ/kg] are:
$w_c=\frac{C_{p_c} T_{0_2}}{\eta _c}\left[\left(\frac{P_{0_3}}{P_{0_2}}\right)^\frac{\gamma_c-1}{\gamma_c}-1\right]$ and $w_t=\eta_t C_{p_h}T_{0_4}\left[1-\left(\frac{P_{0_5}}{P_{0_4}}\right)^\frac{\gamma_h-1}{\gamma_h}\right]$
where the subscripts "c" and "h" refer to cold and hot values, respectively, and $\eta$ is the isentropic efficiency of the compressor or turbine. A piston engine performs the actions of "intake, compression, expansion, exhaust" in the same location at different times, while the gas turbine engine performs these same functions at the same time in different places. Hope this helps.
