How to handle painted backsplash that now has bubbling paint
There is a reason that painted surfaces are generally not used for backsplashes, and you have discovered it.

The standard approach to both counter and backsplash is to have an impervious surface and a seal between them. This is virtually impossible with paint. Occassionaly you will find heavily sealed wood (clear epoxy, marine varnish, etc.) used for either the counter or the backsplash, but they need constant monitoring and repair if scratched or chipped.

You should consider using tile or metal as a backsplash (as suggested by hortstu in his comment). It should also be sealed to the countertop at the base (clear silicone at the bottom edge of the tile/metal).