We agreed that if I help a bit with repayment I will keep the cards with me so he can't use them.

That's called enabling him. Srsly, this guy is old enough to NOT to rely on his gf for this sort of stuff. You're NOT doing him a favour.

that I worry about as his debt is about to go up even more and he doesn't consider it a problem

Of course it's going up. He's not actually working on himself, he's not in touch with a financial advisor, he's not actually learning how to budget.

Other than the financial disagreements, our relationship is healthy and no one has made me happier

Finances are one of the top dealbreakers out there for couples. And the way your SO behaves....most women I know would have run ages ago.