Parent liability for 18 yr old in accident

What is the name of your state (only U.S. law)?
California
Our 18 yr old daughter has been in a couple of accidents already. We are the legal owners of her car and she is on our insurance policy. We want to help her pay for her insurance and keep the cost as low as possible, but do not want to be liable for any additional accidents. What would you advise? Should we transfer the car to her name and keep her on our policy? Or would she need to have her own policy? If we help her pay for her own policy, would we be liable then? If getting her her own policy is too expensive and we decide to keep her on ours, how much coverage should we have? We live in California.

If you own the vehicle, you are liable. Plan to sell the vehicle to her for ~$50.
Being a first time insurance, it will be expensive for the first few months, but after 4 months or so, she should be able to find a better rate with another provider.
She can go insurance shopping before actually owning the vehicle and set the policy up to start the day she plans to change the title over and register the vehicle.
Once the vehicle is in her name and insured, you will no longer be liable....

If you part with all evidence of ownership, then you cannot be held responsible for your daughter’s negligence in operating the vehicle.

But as long as you maintain ownership and she is driving the vehicle with your permission - express or implied - you can be held vicariously liable for death or injury to person property resulting from her negligent use of the vehicle. (California Vehicle Code Section 17150).

However, as the owner your vicarious liability would be “limited to the amount of $15,000 for the death of or injury to one person in any one accident * * * $30,000 for the death of or injury to more than one person in any one accident and $5,000 for damage to property of others in any one accident.(Cal. Veh. Code Section 17151(a)

So it is your call as to whether to transfer the title over to her or secure adequate liability coverage.