Tough Spot in the 1k with a pair + a monster draw

[DJA]

If he is aggressive, he can't have AK, AQ, AJ, AT, or 44 IMO he would 3 bet these hands all day. He may even 3 bet the QJ... He may have AA or QQ and been playing tricky, but this is unlikely. He is giving enough action, he doesn't have to trap.

If you are losing, it would probably be to 1 of the following hands:
QJ
A4
Q4
J4
KT

I would think I am likely ahead here, so what does he have. My bet is weak A, weak Q or flush draw.

.5 to 1.0 pot sized bet on the turn.