You have mold. Mold is a way of life in a humid environment. From the pictures, I'm guessing you have kids. Mold can effect lung development and breathing. Many say that a little mold never killed anyone. I was raised on an island in a moldy house. We left the light bulb on in the closet to prevent mold growth - the ignorant 60s.
After the hurricane we had massive mold - flooding, humid environment, no electricity for a week. I scoured the EPA and FEMA websites. FEMA lady told us to bleach the walls and paint with KILZ.
For an immediate result, I would bleach the painted wall, let it dry and use lysol. Then I would paint with Kilz and apply a semi gloss paint. I would NOT tile over it.
I am attaching a picture. This is a bedroom that we took out the drywall because of mold from the hurricane flooding. This is the backside of our tiled bathroom. Notice the mold spots on the wall? We will be ripping out the tile and the crummy drywall to this bathroom later. We are tackling one project at a time.
I suspect they used plain old drywall (white sheetrock) or green board (green drywall) in your house too. To make it right - I'd rip it out. There are better products. Durarock (cement board) should be used in bathroom showers.