> Suing Americans can be awarded enough money to pay for their medical bills in some cases

That's compensatory damages you will be awarded that in both systems.

The main difference there is that in most European countries medical bills are cheaper, public health care will cover most of the costs and most people are insured for the rest. If the costs are important, your insurance and the insurance of the party at fault will negotiate. They will only involve the justice system if they can't agree.