There are few illustrators who have a more recognizable look (and a longer productive career) than Milton Glaser.
Here's the thing: When he started out, he wasn't THE Milton Glaser. He was some guy hoping for work.
The rule, then, is that you can't give the client what he wants.
You have to give the client work that you want your name on. Work that's part of the arc. Work that reflects your vision, your contribution and your hand.
That makes it really difficult at first. Almost impossible. But if you ignore this rule because the pressure is on, it will never get easier.