Though in my household I can say the door to the small room rarely gets closed and I can't convince anyone else to put the damn lid down. Until slurp-slurp-slurp the dog starts drinking it, in which case there's a "stop drinking from the bowl!" followed by closing the lid. I'm not bitter. I try to choose my battles, and "turn the fan on when you poop" is a hill more worthy of dying on. Still haven't really succeeded with the kids yet.
I hate this type of setup, because it means I have to walk to another room before I can properly wash. Then again, I believe many people only use toilet paper in the US, so they are probably used to it..
Windows also work well.
I don't think I have ever, in my life, been in a bathroom with a "proper" vent fan. AFAICT by observation, regulations must require a fan that makes seems to only make "chugchugchug" noises while seemingly doing nothing to the air.
Good bathroom ventilation is important. Controlling humidity is a big thing, but also properly ventilating things like acetone or hairspray or other things along with bad odors is important.
I do a lot of house repairs. I don't think I've ever seen a bathroom fan that goes outside except in mobile homes. They just vent into the attic. Same with many kitchen fans. They should all go to the outside otherwise they could cause moisture issues in the future.
Where is your toilet?