Answer: What you're seeing is often called "pink mold." It is actually a type of bacteria, Serratia marcescens.
Bubbly liquids often dislodge this growth even when scrubbing does not. Try dribbling club soda along the top edge of the tile, or at least above the area where the grout is pink.
Then quickly follow with an old toothbrush to distribute the soda along the grout lines. Pour on more soda as needed so all of the grout gets wet. Wait a few minutes, and the pink color is likely to disappear.
If it persists, repeat this procedure using hydrogen peroxide, in the three percent concentration sold in brown plastic bottles at drug and grocery stores.
The organism you're dealing with can be deadly in certain circumstances, according to a November 2011 article in Scientific American. It's a top cause of certain hospital infections and was implicated in at least one death in the 1950s when it was sprayed over San Francisco Bay in a test of how bioweapons might disperse.
But on grout and shower curtains, pink mold is really just a cosmetic problem. The bacteria grow on surfaces that stay damp for long periods, especially where there is also soap scum.
So to help prevent regrowth, wipe down shower walls with a towel or well-wrung-out sponge after the last person has showered each day. A squeegee isn't as effective; although it wipes moisture from the tiles, it leaves grout lines damp because they are recessed.