The bathroom fan runs, but the mirror still fogs

The paper test
Hold a single square of toilet paper up to the fan grille with the fan running. If it sticks, the fan is pulling air. If it drifts down, the fan is making noise and nothing else.
Three usual reasons
The grille and the blades are packed with dust. This is the most common and the easiest — pull the grille, clean it, clean the fan. It'll move twice the air it did.
The fan is too small for the room. A lot of builder-grade fans were chosen by price, not by the size of the bathroom. It runs, but it was never going to keep up with a hot shower.
The duct doesn't go outside. This is the one that matters. Some fans were vented straight into the attic. Every shower pumps warm, wet air onto the underside of your roof deck, and that's how you get mold and rot you don't find until it's a roof repair.
What I do
Clean it, test it, and look in the attic to see where the duct goes. If it needs a bigger fan or a proper vent to the outside, I'll tell you what that involves before anything gets ordered.
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