Just a thought:

If he 4-bets the flop but checks the turn, there is a good chance he has a marginal hand. If he was sure he had you beat he would bet here for value, hoping you would raise.

By checking yourself, you pretty much guaranteed you would get bet into on the River. So you paid 1 BB to get to showdown.

BUT, if you had bet the Turn (since he showed weakness by checking) there is a chance he would have mucked it (he may not want to pay 2 BBs to get to showdown if he thinks you are going to bet the River as well.) If he calls, he’ll check the River and you can see the showdown at the same price (1 BB). By betting the Turn, you get to showdown for the same price but get the added chance of winning outright if he folds.

SP