AJ in MP with 20 BBs in MTT - Hand of the Week 1B

AJo in MP w 20bb, on the flop, the hero hits second pair on a draw heavy board. Hero makes c-bet which is small and gives opponent good odds to call if he's on a flush draw. Brick hits turn, hero makes a probe bet on less than 25% of pot. This small bet gives opponent correct odds to call on flush draw and good implied odds to call on straight draw. The probe bet on turn also makes total pot, when opponent calls, greater than hero's remaining stack. Therefore, Hero was pot committing himself with turn bet. I think the turn bet was a mistake. With 20 bb, Hero does not have enough chips to play post flop, especially against opponent who has 40 bb. Hero should have pushed all-in on turn to prevent opponent from having good odds (pot & implied) to call w draws. If opponent had a Q, he probably would have raised Hero's probe bet on turn. Therefore, it is more likely opponent was on a draw and Hero should have played hand faster. With 20bb, Hero did not have enough chips to play post flop.