My insurance said they had done a "public domain" search then asked if my car was turbo charged... Is this a normal question or insurance to ask?

Insurance companies have a database that tells them if you're telling porkies in regards to claims or driving convictions. In reality I think they should stop asking you all the questions as they always search the database to confirm what you've told them.

As for the car though they would only have access to the information you have access to, as in MOT, tax and basic details of make and model etc.

I can't think of anything they can search which will tell them that the car is turbo charged?

It's think it's fairly normal quistion to ask especially if some in a car range are and some are not.

You should look back at your own posts and you'll see why someone might be thinking you have done an engine swap. Have a look at your own profile for recent visitors too