“Bathroom fan” usually means a fan that takes air from the bathroom and moves it outside, i.e., it’s an exhaust fan. A bathroom fan performs two functions: (1) removing unpleasant bathroom odors, and (2) removing humid air from the bathroom. If it’s the first function you want, the typical bathroom fan connected to a duct that leads outside is what you need. But if you want to reduce the humidity in the bathroom, there’s an alternative you might want to consider, especially in the wintertime or perhaps in the cooler months when you’re neither air-conditioning nor heating.
First: It’s a good idea to reduce moisture and humidity levels in the bathroom and running a fan to move air out of the bathroom is a good way to do that. If you take a shower or bath and then do nothing to dry out the bathroom, you’ll eventually have all kinds of problems with mold and mildew, peeling paint, rusting metal, rotting wood, decaying drywall, … and your towels won’t dry between uses. You might also get nasty stuff growing on your shower curtain, but you can wash your shower curtain instead of throwing it away and getting a new one.
In the summertime, it makes good sense to use the bathroom exhaust fan to remove the post-shower warm, humid air from your bathroom and replace it with cool dry air that comes from the rest of your house.
But in the wintertime, your house actually could use that warm humid air. If you move it outside with a bathroom fan, you’re not only getting rid of something useful, but you’re also getting rid of air that you paid to heat with your home’s heating system.
During cold weather, I’ve had good results dehumidifying the bathroom with a desk fan placed on a bathroom shelf with the airflow directed to the open door. This moves the warm humid bathroom air into the hallway, and from there it circulates throughout the rest of the house. As air moves out of the bathroom it is replaced by drier air from outside the bathroom. Some of it comes in from the hallway, as the fan is mixing the bathroom air with the hallway air. But there is also some air movement from the bathroom heating duct of the forced-air system. Houses generally need some additional humidity in the wintertime, and this practice helps out in that regard. It also helps warm the air inside the house by capturing the heat from the hot water. All in all, it seems to work. The bathroom, and everything in it, gets dry. The rest of the house gets a little heat and humidity that it needs.