We want to cb here because there are many worse hands that will call, if we get raised then we need to play some poker and figure out what type of villain were up against and make a decision on how to proceed.

We can check call and pot control with hands like TJ, QJ, 9Ts, TT etc

Yes this pot hits his calling range pretty hard, but with TPTK you are mostly far ahead. So this is a clear cbet.

Turn is probably the worst card to come. It is now a clear check, as betting would scare away only hands that our AJ would beat. when he potts it now, i would fold. I guess up to a half pot bet you could call as he might try to steal the pot with with a worse J. If he bets on the river this hand is clearly over for us

There's actually some merit in taking the c/c line against a thinking opponent, but given the context of an anonymous NL5 game, you should definitely just bet for value. There're plenty of worse hands that could call you and it's unlikely that your opponent will take a dishonest line with a draw so you don't have to worry about potential bluff raises.

As played, your hand is probably underrepresented enough that you could call one bet, but given your lack of position and the absolute worst card in the deck hitting the board, a call would've been marginal at best and spewy against most loose-passive opponents.