Q:Right now I have an auto insurance policy with company X and because I had a few claims already my premiums are increasing. The question is, in case I switch to another company, will my current company share the claim history with my new one?
A: In most cases you should expect that this information will be shared.
Most of the insurers take part in a nation-wide database that is designed to keep track of any claims. They are also able to see your personal MVR records for any violations or accidents you’ve been involved in. So it is quite hard to hide something from your new insurance company.
Moreover, hiding previous claims is in fact a very bad idea. In case you state fewer claims than there are in reality, in case they are found you can instantly get a refusal on a new claim, a significant raise in rates or even canceling of your policy.

If your insurance rates were raised because of claims, you can consider other insurers, but make sure they a fully aware about the claims you have had in the past.