In a pure turbojet pretty much all the thrust is coming from the expansion of gasses from heat released by the burning of fuel in the compressed air stream. The acceleration of the gasses out the burner can and on to the tailpipe nozzle is the primary action/reaction going on that makes thrust (and mostly in the convergent nozzle of the tailpipe).
The thrust force itself is not "pushing" on anything. It's gas that is expanding and has only one way to go, so it accelerates that way and you have your Newtonian action/reaction. Like letting go of a balloon. The thrust comes from the acceleration of air, being compressed by the balloon's rubber, out the ballon's mouth after you let it go.
Thrust loads from the compressor are largely offset by drag loads on the turbine, so the spool is sort of trying to stretch itself all the time, although there may be a net forward thrust component to the extent that the compressor's thrust load exceeds the turbine's drag load, and the ball bearings will be designed to handle whatever axial load there is. I've never found any source that says that any thrust surplus produced by the compressor itself is significant in the total thrust of the engine.
Also think of a turbojet with a centrifugal compressor. No blades making forward lift, just curvy channels flinging air out. How could you get any thrust from that? The centrifugal turbojet is getting its push from the same place as an axial one; between the turbine and exhaust nozzle.
On a turbofan, a significant part of the energy in the exhaust flow is extracted by a separate turbine to make torque to drive the fan. A turbofan is more or less a free-turbine turboprop with an un-geared, fixed pitch propeller with oodles of blades. There's a bit of thrust out the tailpipe from expanding gas energy surplus to what was extracted by the fan's turbine, and perhaps a bit of surplus thrust load from the core compressor, but most is from the fan, and in that case the fan's forward bearings will be designed to handle most of the engine's thrust capability.