3-300 Spread Limit. Blinds are $2-$3. Villain starts with $120. I have him covered.

Villain limps utg+2 and it's folded to me in the Big Blind and I have A J. I raise to 15 and he calls.

Flop is A 9 4 I bet $20 and he calls quickly. Pot is now $70

Turn is 10 What do I do?

I seem to run into these situations a lot at the local casino. It seems like 3 or 4 players are always playing a short stack badly by calling raises preflop and playing passive on the later streets.

I feel like if I lead out for $35 - $50 and he raises, then I am beat and should fold. If he just calls the smallish bet, then I am still stuck if the river is a blank and if I check, then he can easily bluff me off my hand. If I put him all in, he only calls with better. Is there an optimum line of thinking?

In NL, you probably won't be getting 3 streets of value here anyway, I would assume spread is the same in this situation since he is so short. Since you are gonna check anyway, turn is fine.

Against the average low limit player (unfortunately you have not really provided reads for the V), I'm fine with a check/fold here. If he checks back turn, I probably bet/fold the river on the small side (35-40). Note, if the river is another I will lead enough to put him AI (85).

Against most villains I'm check/folding after the Th hits anyway. There is very little that you beat at that point because A9+ beat you, flush draws got there, and T9 beat you. There is much else that should be calling flop. Villain with a short stack shouldn't be calling flop bet with marginal hands hoping to take it away later.

In any case, you can't bet fold. Villain doesn't have enough left, if you bet $40 and they shove your getting better then 4-1 on your money. The outside chance they have a single high heart or have some hand you can draw out on justify calling all but the most nitty and those villains are never betting at all unless they have you beat so you can easily just check/fold against them.

There are very few villains I would continue on such a bad board. Villain would have to have a specific tendency that I'm targeting here. This is actually a more interesting situation if a brick hits on turn, something like 7c. There is a better chance hero is winning then and the right move depends on your knowledge of villain's range. In general though, you should treat this like any passive player. Raise pre for value, c-bet all but the worst flops and then only continue if the situation is good enough for a big hand. In this situation the Th shrinks your hand a lot.

Probably check folding. Agree with what QuadJ is saying, you don't beat anything.