its very simple really: jax only strength is in his counterstrike because that enables him to continuously auto attack and staying out of range is very very hard because of his jump and also flash and if you have enough damage to kill him from range, you will die from one counterstrike easily which means he can always flash kill you.
champions who do best against jax are fighters who can take his counterstrike and then out-dps and/or burst him like darius, garen etc.
its especially easy if jax jumps on them because then not only did he waste his jump so he cant kite them anymore but he also wasted his counterstrike for engaging instead of dodging their attacks at a vital phase of the fight.
for example: jax jumps on garen and he starts uses W E. he will outtrade jax even during the stun and he will then be able to auto Q R freely and most likely instagib jax and still have 40-60% hp depending on the stage of the game (this is at 1 item point).
or jax jumps on darius (1 item point again. jax has rageblade, darius has phage, giants belt and chainvest). darius tries to walk away and gets stunned while jax pummels him and deals some damage. then darius uses auto W and walks away a few steps to Q with blade. he then stays at the same distance because both have similar movementspeed so neither can hit the other but darius cooldowns are very short while jax counterstrike is very long so during this fight, no matter what happens, he will not have a second counterstrike. so jax can either Q again to eat another W Q or he can walk away in which case he gets hooked back, auto attacked and Qed again into R for the kill. darius will probably not go below 50% health from this.
now the real problem for jax in this matchup: if jax wants to kill minions, he has to auto attack them in which case he will eat a free Q from darius. this is inevitable, therefore jax HAS to trade with darius to be able to CS or he will get harrassed down for free which, ofcourse, loses the game instantly. so if he HAS to trade you have to ask: CAN he trade? no, he cant. darius is stronger and he can kite jax and he can heal and he can choose when to re-engage with his hook and he has a mega strong passive for extended trades and an execute. jax has no fucking chance to trade with darius so he has to all-in him but then again, can he win an all-in? if darius wastes cooldowns or misspositions, jax can then all-in him and kill him and snowball because once he is ahead, he CAN outtrade darius but he can ONLY get ahead if darius makes a severe mistake. he cant get ahead from "skill". their kits dont allow it.
this means if you lose a fight with darius against jax you horribly missplayed and you should go to the forums and open a thread about how broken everything is.