My bathroom is very tiny, and it has an exhaust fan however it really doesn't do any good.?

Our bathroom is very small and has no windows it does have a small heater and small exhaust fan however the exhaust fan doesn't really pull it's weight. A ten minute warm shower has the mirror fogged up and steam everywhere. We used to leave the bathroom door open but we have a newborn and the smoke alarm is right beside the bathroom door and goes off immediately! Help I don't want a moldy humid bathroom but I'm out of ideas any tips?