That's a pretty bad case. I think you should figure out what makes a champion good against another one and use that. Irelia does well against Karthus for a lot of reasons: She has a jump, a stun, cc resistance, true damage, and a heap of sustain. The jump allows her to dodge Qs for a long time or to jump to you and begin a fight on her terms. The stun/slow is useful for dueling, which she can and will win. The cc resistance allows her to negate your wall faster. The true damage allows her to build tanky and still do decent amounts of damage, even if you build tanky. And the sustain is to outlast your poke.
A good Irelia could punish you pretty easily. Though, if you're good too, you'll find ways around that. Don't worry too much about who's good against you as what you should do to avoid them punishing you.
Irelia can't really dodge with her Q, it's a good dash attack but is situational in whether it's a good dodge or not, she can't use it to poke in and out like LeBlanc, good engage and great for juking ganks but you need a lot of low health minions (or lots of AD) to really use it as a reliable dodge. The CC I never really had trouble with, either she had more hp and just slowed me and then I slowed her or she had less hp and stunned me but still ate lots of damage from my defile. Also her passive tenacity isn't really noticeable for me unless she is getting ganked.
It's quite probable I faced a crap Irelia (I do face a lot of bad ones as Singed as well) but she didn't bother me at all. The only champ who defeated me was Pantheon because he can poke better then I can though I did make him work for it but his range is just too good but I have trouble with Panth in general because of the damn spear range, I think I will probably buy Panth next. I'm guessing speaking of Gankplank he will be good against Karthus top as well, will just have to Q dodge if I see either of them when Karth top
She can use her Q to dodge, though. Good ones will do it. If you're keeping close to your tower, she'll have problems engaging on you, which it appears to be the case. I'm not a top nor Irelia player, but I'd be getting those Merc Treads and a Wit's End pretty early against you. If you're going tanky, you'll likely be on the lower end of AP, meaning fighting you early with only these will make you B back to base a lot. Of course, every time I make you B, my goal is to reset the lane by pushing the creep wave to your tower fast or to keep it nearby my tower so you have nothing to do top. The closer to my tower, the more likely I am to engage on you. So, eventually, you'll end up with a creep wave hugging your tower, and I'll have nothing to do. Depending on the jungler though, we can continue to make you go B, punishing you ridiculously. If the jungle is bad, I can just gank mid or steal jungle farm. You either push the lane, putting yourself in a bad position again, or you've kept the creep line nearby, and I just continue to hurt other lanes. Again, I'm not a top, so this is just what I'd do theoretically. I'm sure there's better answers.
Simply though, you can never push her out of lane if she's trying. She can push you out of lane whenever she feels like it.