That undersells the turbine. Turbine powers not only the various compressor stages, but also the turbofan itself. All power comes from the turbine and it is is arguably the most challenging part of a jet engine. Turbine blades are tiny and fit in the palm of your hand with fingers closed [1]. Temperatures through the roof, it needs compressor bleed air to keep the turbine blades cool or the blades would melt. I was part of the team that did mechanical design of turbine blades and one of interesting challenges I remember was the trade off between the pressure-side bleed vs. trailing-edge dump techniques of cooling turbine blades. Aero engineers prefer razor thin trailing edge for better efficiency (pressure-side bleed as you can see in the picture). These are tiny holes/slots on the edge of the blade that bleed compressor air from various stages which is at 800F and trying to cool 2200F blade edge. So mechanical engineers (us) would prefer a trailing-edge-dump technique of cooling where we would make the edge thicker and drill the holes at the apex of the edge for better cooling performance. Pressure-side bleed usually won because higher fuel efficiency requirements and pushed us to do other things to cool the blade, even if that requires pulling air from a cooler compressor stage, which impacts efficiency too.
Another fun fact, the whole thing leaks hot gas like a sieve until engine reaches temperature and the tip of the turbine blades seal the hot gas path through elongation along the radial direction. Reminds me of the famous SR-71 fuel tank seals.
Jet engines are insane. So much complexity.
[1] https://www.theengineer.co.uk/media/4kgoz1mg/rr-turbine-blad...