I don't know what European country you are talking about but here is what I did after I moved from the US to Cyprus. I researched what was necessary to transfer my US license to Cyprus. Notice I said transfer - Cyprus allows you to transfer your driver's license from the US. It does not allow transfer from some other countries (like the Philippines) and in that case you have to go through a driving school (that costs money) and take a test.

First, before I left the US, I went to the DMV and told them I lost my license and got a new one.

Then in Cyprus I transferred my US license and got a Cypriot one. I can't remember the cost but it probably wasn't much. I did have to fill out a form and provide a photo. They would not transfer my motorcycle endorsement (that was annoying). They took my US driver's license - the one I had reported lost - and gave me a Cypriot license.

2 years later I get a call from the US embassy telling me they have my US driver's license (the one I turned in to the Cypriot DMV) and asked me if I wanted it back. I went and got it even though I did not need it.

I was able to renew my US DL online when it was about to expire (had it sent to my sister's address in CT).

So now I have a US DL and a Cypriot DL. I don't think I will be able to renew my US one online next time and I can't do it in person because I can't comply with Real ID because I do not live in the US anymore (can't provide the documents they want for REAL ID). I don't want to get a license using my sister's address because I do not want CT to think I live there when I don't and try to tax me there.