Unfortunately you made at least 2 critical mistakes in the hand, the first mistake is not raising pre flop, you start the hand with a little under 14 BB's so it really is player dependent but for most players it would just be a shove all in pre flop move, others might even min raise, bet the flop then shove the turn but never ever limp your hand. Because you limped you hand the player with J2 somehow feels lucky about his hand or that he can out play you having position on you and that could have been avoided by raising or shoving.
As played when it comes to the play on the turn just min raising their raise with your stack size is your second mistake. By min raising like that you're now giving them the odds
to call to try and hit the straight. Instead on the turn you should put all your chips in as the re raise although I feel they still would have called