When Blighted Battleaxe deals 4 damage, is it dealt this turn or the next turn?

The damage is dealt this turn — during the Ending Step of your own turn — and it's gone before the next turn starts.

Here's the timing: Blighted Battleaxe's trigger fires "at the end of your turn" (Ending Step), unattaching and dealing 4 to the equipped unit. Then, still in that same turn, the Expiration Step's Ending Special Cleanup heals all units (Rule 317.2.b), wiping the 4 damage. So the opponent starts their turn with a clean unit — the damage never carries over (FAQ 11352).