Being sued for auto-accident

2 years ago, I was driving my mother in law’s car to work, the car is now legally mine but on the date of this accident, my name wasn’t on her insurance.

I was making left on a T-type intersection as my light was green. Directly in front of where I’m coming from, there’s a dispensary and this young lady peels out of the driveway of this said store. She made a quick left and crossed me but by the time I could break, I wasn’t able to stop in time and hit her left back wheel and fender.

This is her fault as I was already turning on my green light and she should not have been making a left turn since there are double lines.

My question is, am I screwed here due to the insurance policy not having my name on it? even though I should be able to prove it was her fault??

Thanks in advance for any input.