What happens at the end of a players turn to a buffed creature that took more than its unbuffed base health?
Your Blood Bairn will survive.

This is because "Until end of turn" buffs and Damage are removed at the same time during the Cleanup-step.


  514.2. Second, the following actions happen simultaneously: all damage marked on permanents (including phased-out permanents) is removed and all “until end of turn” and “this turn” effects end. This turn-based action doesn’t use the stack.