If my child was injured when she turned a 4-wheeler over on herself, can I hold the ATV owner responsible for her medical bills even though I allowed het to operate the it?

The owner of the ATV is liable for your child's injuries.  If the ATV owner has an applicable insurance policy, your child's personal injury claim should be filed with that insurance carrier.  Notify the insurance carrier in writing of your child's personal injury claim.  The insurance may be on the ATV itself or it could be homeowner's insurance if the accident occurred on the ATV owner's property.

When your child completes his/her medical treatment and is released by the doctor or is declared to be permanent and stationary which means having reached a point in his/her medical treatment where no further improvement is anticipated, obtain your child's medical bills and medical reports. 

Compensation for the medical bills is straight reimbursement.  The medical reports document the nature and extent of the injury and will be used to determine compensation for pain and suffering which is an amount in addition to the medical bills.

If the case is settled with the ATV owner's insurance carrier, NO lawsuit is filed.

If you are dissatisfied with settlement offers from the insurance carrier, reject the settlement offers and file a lawsuit against the ATV owner for premises liability if this occurred on the ATV owner's property.  You will need to be appointed guardian ad litem to file a lawsuit on behalf of your child because your child is a minor.

If the case is NOT settled, your lawsuit against the ATV owner must be filed prior to the expiration of the applicable statute of limitations or your child will lose his/her rights in the matter forever.

If the ATV owner does not have insurance, you will need to sue the ATV owner on behalf of your child.  If you get a judgment against the ATV owner, you can enforce the judgment with a wage garnishment or a lien on his/her property.