I would write (letter/email) the landlord, explaining the problem, and formally request he fix the fan before mold gets out of hand. Mention your health problems and how you'd like it not to become an issue.
In the meantime, have you tried setting up a box fan in the doorway? Should help some

Also, try not to take a super long or steamy shower and when you do shower if at all possible, do it with the door open. This allows the moisture to travel outside of the bathroom.

QueenOfTheFall also makes a very good point about contacting your landlord. Mold is certainly a health code violation!

Invest in a dehumidifier.

I suggest getting a small floor heater. There are several models that run on a timer. Just set it to twenty minutes, hopefully this will help dry the bathroom out. I use this type of heater in my bathroom and it really helps with the moisture

tell you landlord to fix the exhaust fan. If mold sets into the ceilng he'll have to spend a lot more money remedying it than if he just fixed the exhaust fan.

I have the same issue and I also use the open door/space heater solution but I live alone so open doors are easier for me.

Check the codes in your area but most places I've lived in require either a window in the bathroom or an exhaust fan. Otherwise this is a violation. I also have the world's tiniest bathroom with a srawny window that offers little relief to the moisture build up. I often leave the door open and have the ceiling fan in the hallway on. This helps a lot. Try placing a fan in the doorway of your bathroom to help dry it out after taking a shower. Also leave the door open the majority of the time when you are not at home.

srawny=scrawny

I've heard if you keep an old towel in the bathroom that you wipe down the shower after your done with it can really help. Obviously you'll need other tips, but this might help in the mean time.

Disconnecting the fan is a completely unacceptable solution to that problem. He needs to fix it. Like now.

The landlord had a fan but you didn't like it. He is under no obligation to replace it with something more to your taste. I understand your issue but that doesn't change his obligations under the law.

If your lease is up soon then move. If you're staying then DIY a replacement. Fussing about the issue won't get you any relief.

You can get a super quiet Broan exhaust fan from home depot for about $160. Slightly noisier ones are far less. All will be more efficient in terms of air movement and electric usage and certainly won't rattle. Just look for one that's approximately the same sized opening in the ceiling.

If you can't afford to have an electrician come out and install it go to your library and check out a book about home electric wiring and install yourself. It is not hard.

I don't know if you requested the fan turned off of the landlord turned it off due to neighbors complaining. If you did not request the fan turned off, your landlord is definitely under obligation to provide you with a new one. If you requested it, have it turned back on - if the rattling is a sign of a problem with the fan have the landlord fix it, if not just live with it.

Most places have extremely strict rules about mold and mildew to the point that were the problem to persist even with the fan, you can require the landlord to both buy you a de-humidifier and pro-rate your rent for increased utility bills if you pay your own electricity.

Read the post again, linbo- the renter didn't turn off the fan, the landlord did. Which is actually probably not legal in some states! Depends on where you live. Elizabeth, look up the rental laws in your state and town. Especially look up anything on mold.

I had exactly the same problem when I was living in a basement apartment. To the point where there was black mold growing like a carpet on the back of the toilet when I found it. There are very few really satisfying solutions to this kind of problem. The best one is a fan, so don't give up on that option until you've waved every law you can find under your landlord's nose.

Dehumidifiers are just too big for the majority of apartments. If you do worry about mold, one line of defense is an anti-mold spray. It doesn't work as well as just drying the place out, but every bit helps.

Moisture absorbing crystals you can buy at places like Home Depot work a little, but you have to replace them often.

Leaving the door open while you shower helps a lot, even more so if you have a little fan you can stick in the doorway. Getting rid of excess water on the walls with a towel or squeegee really helps too, although of course, you have to hang the towel somewhere else to dry.