Misty windows suggest the room is humid, which will cause condensation. This might be due to not opening the windows, or there could be another cause, possibly related to moisture from the bathroom.
I can't make out what size the fan is, but I expect it will be 100mm diameter. It looks like it is in the ceiling and blows steam into the loft.
It is very important in that case, that the steam passes into a sealed duct and passes out through a wall or eaves to escape outside the house. If the duct leaks, steam will escape into the loft and cause excessive condensation. If the duct is not well insulated, or the flexible type, condensation is liable to collect inside it and leak out.
If you're refitting the bathroom, and the fan is the ceiling type, I strongly recommend an inline ducted fan, which can be much more powerful, and quitter, than the old one, and new rigid ducting laid with a slight fall so that any condensation drips outside the house. Wrapping the duct in mineral wool quilt will minimise condensation.
This will be more expensive than flexible duct, but a better job. A plumber can probably do it because the duct is very like plastic soil-pipe. It may mean crawling about in a dusty loft though.
This is an example of a ducted fan. The one I show is a good make, there are cheaper ones. It has two or three times the power of the sort builders usually fit in bathrooms, and should be wired to the bathroom light so it comes on with the light switch, and runs on with a timer. Cost of electricity, and noise, are trivial.