50nl AQ TPTK facing flop raise and turn bet

[Keith]

villain is 18/14 opening 43 on the button
Poker Stars, $0.25/$0.50 No Limit Hold'em Cash, 5 Players
Poker Tools Powered By Holdem Manager - The Ultimate Poker Software Suite.
BTN: $56.22 (112.4 bb)
SB: $55.95 (111.9 bb)
Hero (BB): $97.63 (195.3 bb)
MP: $129.31 (258.6 bb)
CO: $44.34 (88.7 bb)
Preflop: Hero is BB with Q A
2 folds, BTN raises to $1, SB folds, Hero raises to $3, BTN calls $2
Flop: ($6.25) 2 3 Q (2 players)
Hero bets $4.50, BTN raises to $12.47, Hero calls $7.97
Turn: ($31.19) 7 (2 players)
Hero checks, BTN bets $15.34, Hero ???
felt his range was weak with him min raising on the button . Possibly should have made raise pre $4.Thought he could have lots of Ax, suited connectors, mid pocket pairs in his range. Thought with his flop raise he could have been trying to push me off any hands where i missed the flop , possibly where he had draws giving him equity i.e flush draws , 56 A4s A5s etc
Should I have shoved over on the flop? I didn't think so since his flush draws i was giving him roughly 50% equity and only hands i'm really crushing are KQ QT QJ. what to do on the turn , having shown weakness on the flop , couldn't see any point leading, but then again ...would bet folding $15 get him to fold his flush draws and gutshots but by then half an effective stack is in the middle and i'm pretty much committed. betting small lets him hit his draws cheaply or shove over.