I like your preflop sizing, most people 3bet too big there.
1/2 - 1/2 - 1/2 is a good sizing plan for a dry board
This flop is very dry, no draws except rare gutshots and 2overcard (one of which is often dominated), so going with a 1/2 three times plan is good.
Turn, sudden it gets incredibly wet. That card hit both of your ranges: you can have any Qx, any draw (fd, AK, AT, KT, T9), and he can have Qx (AQ) as well as some turned draws (AThh,A4hh,A2hh,Jxhh,and maybe floaty KT, 76hh).
As such, betting 1/2 with 1/2 behind is a bit silly. If you were to bet $50 on the turn with AK, and he were to jam, you will have to call it off. Same for any other 2barreling semi bluff. And usually calling allin with a hand without showdown value is a sub-optimal play. So what other options do you have?
1) bet really small, like $30. That way, you can still bet/fold your AT AK T9.
2) check/raise, both invulnerable value hands and good draws
3) just go allin for 1.5pot with both value and good draws
I prefer option 3), it plays out well. On occasion Ill also bet $30 or check raise. I think JJ is a very good hand to check, for instance. And 76 is a good hand to bet $30 with.
So yeah, your sizing plan is good, but on this very turn card I think I would exit the old sizing plan for a new one.