How do you handle Irelia?

Counter her with Riven or Renekton GGNORE.

bully her early and don't fight her 1 on 1 once she gets that mid game sheen power spike.

I would say Garen, not a direct counter but he's easy and fares decently against her. If you main top and know how to play fighters rush ghost blade but most likely rush sunfire into warmogs and she will lose. The way to play it is pressure her pre 6 then after just chill and farm waiting for her first move. You can take bad trades (don't suicide into her creep wave) because she is weakish early and will go oom quickly but post 6 and with items she will win if she gets stun while you tank minions so wait for her to aggressive.

Alternatively if you are good with/more comfortable on Kha he does probably the best but you need to know how to play him. On Kha play back and farm (Kha does Irelia's job better later the game goes on), if she jumps on you you will win with iso damage and can win duels by evolving r first in order to avoid hiten style.

Thing is Irelia is pretty hard to counter, I mean Jax is listed but she can often best him if he makes a tiny mistake.

Beyond her initial short range dash she has no engage. Someone with hard peel like Nautilus or Alistair or consistent CC like Singed or Skarner can prevent her from going where she needs to be. Then your Marksman can leisurely stay back and shoot her to death.