Aim to trade when Jax has used his E as it will block basic attacks. It has a long cooldown, so make sure you exploit his weakness.
Jax has no build-in sustain and is melee. If you have the range advantage, try to poke and harass him as much as possible to prevent him from looking for the all-in. This will also force him to burn through his potions quickly.
After level 6, every third auto will deal extra magic damage. Do not commit to extended trades with him- especially if he’s been able to stack his auto attacks on minions beforehand.
If Jax decides to team fight, try and lock him down with CC before he has the chance to engage. While his E will block basic attacks, he is vulnerable to disengage and CC.
Jax will be looking to split push in the later stages of the game. Always keep an eye on where he is at any given time. Do not let him take objectives and open up your base for free.
Forcing a fight with a numbers advantage when Jax is split pushing can be a good way to win a team fight. Keep an eye on him when fighting and be prepared to retreat if he leaves the side lane.
Jax’s Ultimate R will give him increased defensive stats in a trade. It is very good for duelling.
Once Jax gets his first item, he becomes much harder to duel unless you have items that match his damage. Don't over commit to fighting him without a backup plan.
As the game goes along, Jax will get stronger and stronger. Keep an eye on him at all times as he will look to split push and take objectives quickly.