What you have is a mud bed on wire lathe. Yes, its a pain to remove but achievable. Wear leather gloves so you don't tear your hands up.
A pry bar and a hammer should get it out. It helps to use the studs for leverage. Keep an eye out on the other side of the walls for damage to the drywall or plaster.
Once you have removed the tile bed, nail some shims (maybe some 1x2's) to the studs to bring the walls out where you want them and nail the cement backerboard up.
If it's a shower install the pan liner first.
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