My take on the situation:
No matter what you do, unless you go out of your way to avoid it, you WILL touch some dirty
bathroom fixture before you leave.
People go to the washroom.. then, they turn on the taps with their
dirty hands, wash them, and then touch the dirty taps again. They touch the lever on the
paper towel dispenser with their dirty hands (unless it's a dryer, some of which have buttons to press), and then the handle of the door
on the way out.
You can't win.
I like to make a game of it. Try as hard as I can to not touch
anything, turn taps on and off with paper towels, have someone else touch them for me. Sometimes I even
wash the taps! It's kind of fun, and you can involve others in your little game as well.
Hint: try opening the door with the
sleeve of your jacket over your hand.