'mmm, that's quite a good one.
Like this though yours may be an older model.
If working correctly (not clogged with dust, fluff, birds nests) it should be pretty effective. You can try holding a bit of tissue, or a smouldering joss stick or fag, close to the grille to see if it gets sucked. Also close the bathroom door and check for a draught being pulled under it.
They can be taken off for cleaning. I presume the outside grille will be difficult to see, but you might be able to peep through looking for daylight.
As it is very cheap to run, you could try leaving it on permanently for a few days and see if that makes as improvement. As long as you have energy-saving or LED lighting (assuming it turns on with the lightswitch) the cost will be negligible.
There is an adjusting screw under the cover, you can increase the run on time to (I think) 30 mins max. There may be a plastic screwdriver-like tool clipped to the inside casing to fit.
You can wash the outer casing and grill, as it detaches, and use a small soft paintbrush on the propellor (after turning off the electricity) if it is fluffy. IIRC the locking screw to open the case is on the bottom edge.
i'm not keen on the ones with a humidity sensor, they can be slow to start and do not detect odours.