Simple physics !!
If you have a bathroom that is almost hermetically closed while you use the bathroom, you have very little airflow, so the steam *hangs around*
You MUST allow for airflow and that can only be done if you make a vent where the air can come into the bathroom and then gets sucked out of it by the fan.
Try taking a hot shower with the door cracked open one day when nobody is there to *bother* you ...
If still steamy windows, then not enough airflow from the fan !
Get a bigger one, or your outlet of the fan is very inefficient !
Well, that is quite simple. Since you are bathing with hot
water, the fog is the condensation of vapour as it touches the cooler surface
of the mirror. If you run cold water beforehand, you will cool down the bath
tub and everything around and the vapour from the hot shower will condense
right there without reaching the mirror.