So your motion-activated fan will go off after you leave the room, and possibly while you are in the bath if not active.
Does it have a run-on time delay?
Can you show me a photo of it?
Can you see the external grille where the steam is supposed to come out? Do the flaps open when the fan starts?
You mention smell of wet cement. Wet brickwork smells different to wet plaster, can you tell the difference?
Wet brickwork is more likely a leak.
Wet plaster might be condensation on the ceiling, often because it is not well-insulated above. Can you look in the loft? Sloping ceilings are often badly insulated.
If you are using a shower, water might be leaking round unsealed gaps, often where the tray, bath or enclosure meets the wall.
How old is the house?