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Villain (BB) 20+bb
Hero (SB) 20bb deep AQ

blinds 0.5/1bb

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Preflop:

Hero raises to 2bb, Villain calls 1bb

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Flop: AT5 (4bb)

Villain bets 1bb, Hero raises to 4bb, Villain calls 3bb

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Turn: AT53 (12bb)

Villain checks, Hero bets.....?

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Hero has 14 bb left on turn.

Villain is a reg. When he donks the flop and call a raise, I think he has 5, T or some kind of draw most of the time.

This is pretty common situation for me. I'm on turn with good hand but I have a little bit more than pot left in my stack. What should be our bet sizing? If I shove, I think that any hand weaker than A is folding. If I bet 1/2 pot, I give good odds to call with a draw and when draw hits in river, its pretty hard to fold top pair because I have only 8bb left...

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+1

You can choose what stack to pot ratio you get to the turn with with your flop raise size. If you don't want to overbet turn for value, then raise flop bigger

If I raise flop to 6bb i have still almost potsize bet on turn (14 to 16). This is not big different situation. Draws still dont get the odds and are folding to shove and I think T and 5 are also folding. What hands are you getting value from if shove turn?

It vastly depends what range you expect villain to flat with. You seem to have reads on him. If you think he has a lot of strong draws and some Ax as well as his T and 5 then shove the bit less than potsize bet, and if he has no Ax/2pair you can probably shove here with a ton of your air as well, especially stuff which has blockers to the flushes hitting.

If you think he has a lot of weak draws and 1-pair holdings and will herofold them, then underbet turn underbet riv is possible, assuming you don't think he's capable of bluffraising.

The post above wasn't really meant to be advice on this specific hand, but rather with the general problem you mentioned of getting to the turn with stacksizes you find akward. If this is happening, you need to focus more on planning what turn play will look like when you bet or raise flop.

I don't mind shoving here. Villain might put me on a flush draw and call with a T or 5 or eg KdJd.
If I have a good read on villain's range and tendencies it might be correct to bet smaller on the turn and than herofold the river if draw hits and villain bets.