How do you remove the old paint in this situation?

Exactly what are you painting?

If it's wood then use a heat gun to remove the existing paint. Heat
guns are good for removing paint from wood. If it's drywall or plaster
I'd scuff it up the existing paint with sandpaper then prime and paint.
That's if the existing paint is adhering well. Bathroom paint is
often problematic and requiring complete removal but if it's adhering
good then the scuff and prime and paint methind should work well.