Will it cause problems to shower without the exhaust fan and only turn it on afterwards?
Condensing water is never good. The vapor will creep in places where it may not quickly evaporate once it has condensed. It will condense easily but dry slowly in cold places and badly ventilated corners (like around the window) and on a cold floor. That means that these places stay moist for longer than one would think. Take a "finger test" in such places an hour after you have finished.

If water vapor condenses between layers like under wallpaper, window paint, between plywood layers or similar, it will start to do progressive damage there, even if it's only present for an hour each day.

Getting the vapor out immediately is much better. It will prevent some of the condensation from happening in the first place, which is much preferable.