This is not an answer. This is only my attempt to organise my thoughts and try to explain it using (simplistic) physical principles (and in the meantime improve my own understanding). All constructive criticism (even accompanied by downvoting) is welcome.
I suspect what you mainly have in mind is something like the following image.
figure: Jet engine Turbomachinery stages (source: A python approach to multi-code simulations: CHIMPS)
Basically the compression stage is responsible for increasing the air pressure (which means that its pulling more air from the surrounding environment and compressing it). The combustor stage is responsible for igniting and expanding it, and the turbine stage is responsible for converting some of the energy to rotation to power up --mainly-- the compressor stage.
The way I understand it is that although the combustor indeed releases the fuel's thermal energy and increase the volume and temperature of the flue gases (not entirely sure the term flue applies in this context here), the pressure that the combustion takes place is atmospheric. So the way I understand it is, that if the turbine stage was not there, then the temperature would have been significantly lower and probably the pressure also).
So, the turbine stage is responsible for keeping the pressure at the exit stage of the combustor so high, because its putting a barrier (which is required to extract the energy).
Obviously, I expect that if the turbine stage is not properly designed (if flow is too much restricted) then there could be a pressure buildup.
So although the compressed air comes in and there is combustion, because its an open process to the atmospheric pressure there can be no further build up of pressure.