U-turn is "U-turn" when listed as a TM in your Bag or when listed on any Pokémon's move-set. However, in-battle, U-turn is for some reason changed to "U-Turn". Since we're using the TM listed or move-set versions of U-turn, we should, and do use "U-turn" as the correct form. If we want Shoddy Battle 2 to have 100% correct battle logs, it should be "U-Turn" there, but since that's still in development, doesn't need to be discussed here, and quite frankly hardly makes a difference, it doesn't matter.
U-turn is correct, use it.