Jack lower kicker is one of the only hands you can beat here. Nobody plays J8 or lower and so JT and J9 could really be the only hands you beat here. I don't think he would call when the second jack hit the board with a lower pair. JT is really the best hand you can hope for, but I think more often they will have AJ or JK.

At what specific point did you play this wrong? The guy minraises him back on the flop to make sure he has the goods. Would reraising there be correct? That seems overly aggressive. Folding to any sign of aggression seems wrong. I would have check/folded the turn had the flush draw not come out. Let this guy be overly aggressive with TP Weak Kicker. If he plays like that all the time he's going to lose money with hands like that.