As a diehard Jax player and someone who considers him my main, It took me FOREVER to get as skilled with him as I am and I still can get shut out and shut down pretty hard.
Timing his W to get more hits in, knowing when to use his E for the dodge or just activate it right before a leap for the stun, and knowing whether to level leap first instead of W to keep up with a squirrelly laning opponent are all the tricks of the trade.
Jax can be an offtank without itemizing for it pretty well, especially when you can activate his ultimate for the gigantic bonuses to armor and MR.
Unfortunately, CC, kiting, and well-timed burst damage can shut out even the best of Jax players. You have to find ways to keep your enemy from escaping so you can get in those consecutive hits, keep you passive stacking, and land those third-hit ultimates.
On Twisted Treeline, rush Merc Treads and a Blade of the Ruined King ALWAYS. The lifesteal on the Blade's active is invaluable and super strong, and the merc treads will get you back to stacking passive faster. Generally level Q first or Q and W about the same unless you're not dealing with particularly high-mobility champs.
After Merc Treads and Blade, get a Trinity Force, Gunblade, and some tank items like a GA or Thornmail and Force of Nature. You'll be super fast, super tanky, and super high damage.
Edit: Oh, and that's why Jax is always banned. Ultra-bruiser best 1v1 champ.