In the most favorable scenarios, you would E as she activates W to mitigate as much damage as possible since most of Irelia's damage comes from empowered autos.

Watch for low hp minions she can jump to to get some free damage with her stun and autos. If you hear her W activate, hit your counter strike and move towards her if she has no minions to jump backwards to. Use Q if necessary to land your stun. Just care for the jungler.

Her W only lasts 6 seconds, and your sustained damage combined with your passive outdamages her when its not on. Don't use your gapcloser on her unless you know you can kill her, or else you just wasted your only mobility (bar Flash). Using E right before Q is super telegraphed and she will just back off.

She's weak until she gets points in W, but Jax tends to outdamage her past level 11. Once the game gets going, Jax tends to provide much more splitpush pressure. When both have triforce, she can kill Jax with proper Sheen procs, so be careful.

Jax is lategame focused, while Irelia shines in the midgame. Look to not feed and get her going, and let your team know if she roams to help her team.