It really depends on the assassin (Leblanc, for instance, relies on her W for waveclear if she chooses to do so. Plus she uses mana, so she's not the strongest laner, and she's still a mage so her killing and laning aren't exactly like a pure assassin's).
It's not that they're supposed to be bad, but they're supposed to have a weakness because they excel at what they do best: killing. Mages aren't assassins, they can't always 100-0 people as effectively as an assassin, so their laning is slightly strong. It's why people gripe so much about Zed: due to his energy resource, his decent waveclear and harass, and his passive that works on minions, he's got a pretty decent laning phase compared to other assassins, but he can still assassinate squishy targets then get away with his shadows to compete with the best. That's why people don't like him, because of his lack of discernible weakness.
Fizz and Kat are examples of weak laning, Yasuo and Talon aren't the hottest either but don't have it as bad. I believe Fizz and Kat do a bit more damage as a result, but I'm not entirely sure.
But yeah, laning from what I recall is supposed to be an assassin's weakness. If you remove ignite from the equation, and assume both players play relatively safe, when does a Katarina's kill pressure kick into high gear? A Zed's? Yasuo's? Hell, even Talon but I don't really play him?
Level 6.
Their weakness is getting there, but once they do... massive powerspike.
But I'm neither balancing team nor Master Division, so don't treat my word like law. Just saying assassin's are supposed to have weaknesses beyond just cc, and it's generally laning. And I believe bruisers counter Zed.