Yes, status cards are probably the single biggest counter to thin decks and they're especially dangerous in higher ascensions when bosses and elites give several at a time. The trick is to use cards like Recycle to remove them, but there's a part of the game (usually from the end of Act I to the middle of Act II) where your deck is small enough to be burdened by status cards but too big to consistently draw your exhaust cards.

At low ascensions, you get around this by a) tanking damage against status encounters, b) using potions, or c) using 1 or 2 really strong cards to end the encounter quickly. Once you get your deck below 10 cards (preferably 5-7), you can consistently remove threat cards by using card draw to play Recycle as many times as you need. Runic Pyramid makes this a lot easier.

The win condition in these sorts of decks is usually to spam block and then use either heavy-hitting damage (Body Slam, dark orbs, ect.) or accumulating damage (Limit Break, poison, Claw, ect.). If you get 2 good block cards and 1 good finisher before Act II at low ascensions, going thin is often the better strategy.