Bleach in a 50-50 mix of water is what's normally used to remove mold, and bring back the white. Of course, the more bleach you use in the mix, the whiter it gets (and the more noxious it gets too!)
The end-all solution is to eliminate the causes of the mold. It needs moisture and temperature to grow. Simply opening the door, and leaving it open, just a little during a shower is all you might need. Or, opening the TOP portion of a window a 1/2" or so too along with the door being open a little. If air can 'push' the moisture and heat out of the room, you've dramatically reduced the conditions for mold to grow.
A fan to the outside is good too but I recommend it as a last resort. And, there are right and wrong ways to install one too. If you end up going this route, we'll be happy to offer Do's and Dont's to you even if a contractor is doing the job.
My wife's bathroom is 4' x 8'. SHe takes 15 minute showers. She leaves the door open about 4-5". The window is closed. When she's through, she 'shake's the curtains to get standing water off as best she can. Then, she leaves the shower curtains open for a while while she does stuff like makeup and dressing. Then, she draws the curtains closed and is done! In short, all she's done is allow the heat and moisture to get out of the room. I suggested she even go as far as drawing the curtains closed (as she does now) but leaving a few inches on BOTH ends 'open' a bit to allow circulation. What little mold she gets would be considered normal and is easily cleaned during her regular bathroom cleaning. (She can't be bothered with my 'picky' suggestion. Well, it's HER bathroom.)
Other than that, there's not much else. My best to ya and hope this helps.