In the USA there are tort lawyers that specialize in such claims. The trick is to find one that is a good negotiator and one that is not going to want a big payday for themselves. Worthwhile to write down the situation as best you can and then send it to multiple attorneys and evaluate their responses.
The attorneys I have worked with were not good negotiators and I had to drive the end result. Many attorneys will bill you regardless of the results and they will bill at their hourly rate and 15 minutes is rolled up to an hour and they charge at that rate even when they have clerical staff do the typing or copying of filing at the courthouse.
Insurance companies have literally thousands of attorneys in their employ and it is very much a David and Goliath situation. I had a $14,000 claim a few years ago for damage to my property and the agent for the insurance company offered me $7,000. I told the agent that if I had to go to court it would be for the amount of the loss in market value to my house or more than $50,000 and two days later they agreed to the $14,000 payment. They expected me to settle for the $7,000 as evidently most of the time people do.