I just had my first loss as 
in ranked after going undefeated on him for 8 games. Main issue was having no control because of this piece of trash 
. If I outtrade her she can stun me, if I she outtrades me she can still burst me, and she has better sustain. How do I deal with it? She rushed triforce into a QSS for laning in case you're wondering.
I think your biggest issue was that you bought a Zeal on your first back while Irelia bought Sheen. This gave her significantly higher burst damage in trades and all-in fights, which she could easily take advantage of.
I would have recommended getting Phage (for damage, durability, and sticking potential), Sheen (for higher burst damage in trades and mana), or Bilgewater Cutlass (for damage, sustain, and the slow) on your first back; Zeal was probably your worst option.
I would recommend using your counterstrike defensively and taking advantage of your high upfront burst damage. After level 6 you should ready a passive proc, jump on Irelia, hit her with an autoattack (which should apply Sheen, W, and R-passive), and then back off. If Irelia tries to return damage, activate counter-strike and keep fighting her since you should be within your minions. I would recommend saving your counterstrike for when she activates Hiten-Style to negate her true damage (I forget if it prevents the healing).
Play 
and kill her at level 3 or at least force her back to the point where she'll be 12 cs behind and a level under you.
Yep it was most definitely the Zeal that messed you up as Endless said. I usually have a hard time against Jax top lane as Irelia usually because he takes Sheen or Phage as a first item. Was scratching my head wonder why you were losing until I checked the match history.
In the Jax vs Irelia lane it is all about not blowing your stun before the other person does. As jax you shouldn't really look for trades, you are better off waiting to outscale her.
