If I have Blighted Battleaxe on a body Rengar, I haven't conquered with it this turn, does the damage last into my enemy's turn?

No, the damage does not persist to the opponent's turn.

Here's why: Blighted Battleaxe's trigger ("at the end of your turn") resolves during the Ending Step, dealing 4 damage to the equipped Rengar. Then, during the Expiration Step of the same turn, the Ending Special Cleanup occurs — which includes "Heal all Units" (Rule 317.2.b). That healing clears all damage from Rengar before your turn ends, so the opponent starts their turn with a clean, undamaged unit.