Preflop
Standard raise.
Flop
Standard C-bet.
Turn
You only started the hand with around $80 or 40BB effective, and you are playing, what I presume is a soft 1/2 live game. If these assumptions are true, you are leaving a ton of money on the table by checking back a hand this strong. The hands, you would want to check back for pot control, would be something like JT, where now you basically have the worst kicker, you will ever have. But AJ is easily strong enough to play for stacks 40BB deep, so you should bet around $15-20 and set it up for a river jam. If he check-jam on you, then you beat him into the pot and hope to run good.
River
Something is wrong in the hand history, because you say, a flush is possible, but there are not 3 cards with the same suit on the board. This matter, because we need to know, exactly what the cards are to determine the chance, he has a flush, and if no flush is possible, then obviously there are less hands, you lose to. If there were 3 clubs on the board rather than 2, then you having Jc is quite important, because it makes it impossible for him to have flopped top pair with the backdoor.
As for straights it seems very unlikely, T8 or 86 with no flushdraw would call the flop, since these hands flopped absolutely nothing other than a backdoor straightdraw. Since its a BB defend, I guess, he could have 63s, but there are only 4 combos of that. So all in all there are not that many combos, that got there on the river, as you might think. But of course he could also have flopped or turned it with hands like J9, 44 or 55. Especially J9 makes complete sense for him to play this way, whereas 44 and 55 will check-raise the flop at least some percentage of the time.
At the end of the day you have massively underrepresented your hand by checking back turn, and your plan should have been to play for stacks by betting yourself. So as played I think, you have to close your eyes and call, unless the opponent is an OMC (old man coffee) type. It is close though, because hands like KJ or QJ probably dont overbet for 2X the size of the pot, so by playing passive on turn you have degraded your hand to a bluff catcher and made the river decision much more difficult, than it should have been.