Here is one possibility.
Make a filler/side-splash as shown below. The difficult part is how to fasten it, but the sides-plash portion gives you a way to put a couple of screws into the studs.
I envision the three pieces shown below all glued and screwed together before installation. To look proper, it should be finished in high pressure laminate (Formica or equiv). If you have not done it before, its really not difficult. Lots of Youtube videos to help you out. Just that one piece, on the side-splash covering the screw holes will have to be adhered after installation in the bathroom (to cover the mounting screws).
After you become an expert at applying laminate, you might decide to make a matching backsplash piece, since you will have lots of left-over laminate (as far as I have seen, laminate is only sold in 4x8 sheets or bigger).