Irelia's entire kit revolves around duelling power, and dealing high single target damage to blow up squishies. Karthus has one of the weakest laning phases in the game, and his entire kit revolves around massive late game aoe damage in teamfights. Factor in that you for some reason decided to take heal ignite, and it's pretty obvious that Irelia is going to have an advantage in a 1v1 situation, despite being behind in gold. From a balance perspective it makes sense for different champions to have different strengths and weaknesses.
Feel free to ignore this second paragraph if you don't want any advice on the specific match-up, but here goes. Irelia is a champion with a really weak early laning phase but has a couple huge power spikes. The one you most likely got cheesed by was her level 6 sheen burst, which is ENORMOUS, especially vs an opponent with no armour. Taking heal ignite on Karthus means you have no mobility other than the momentary speed boost from heal, which isn't ideal since Irelia can easily q to a low health minion near you, blow you up and stay on you with her phage passive and another q. IMO the best summoner spells to take on Karthus in virtually any match-up is flash exhaust (In some cases teleport can be subbed for exhaust if you want more map presence), for extra mobility and much-needed duelling power. Exhaust is a really diverse summoner spell that can be used defensively AND offensively, synergizing perfectly with your Wall of Pain to hugely reduce their MR and movespeed to let you land qs basically with your eyes closed and watch each one take out a quarter of your opponent's health. Flash is an absolute must on Karthus for so many reasons, everything from escaping a nasty situation to repositioning for maximum aoe damage in a teamfight. This 1v5 Karthus penta which I masturbate to on a daily basis, for example, couldn't have been accomplished without flash! https://www.youtube.com/watch?v=8KfG7KlOVaY
But yeah Irelia in particular you always need to respect her burst and Morgana snare-esque stun. The best thing you can do to stay safe in lane vs her is keeping an eye on friendly minion's health bars and denying her the chance to gapclose on you by getting a Bladesurge reset on one nearby.