I was in a bike accident where the lady that hit me made an illegal turn (right turn only & she made a left) - is it possible to make her pay my attorney's fees & cost, hospital bill, etc. on top of the settlement offer? I believe she has coverage of $10,000 and that's what they offered me but I want her to pay for everything else as well. Is this possible?

If you have a police report that shows she was at fault, you can certainly sue her. Though if you accept a settlement that may be giving up your right to a suit.

If she only had $10K in liability coverage, then that's all her insurance is required to pay. So to get anything more you would have to sue her personally. If she has no other assets, could be hard to collect.