Not in most cases. According to this estimate, who comes from an organization that is against US citizens abroad having to file[0], 77% have AGI "well below" the cutoff where you would have to pay taxes. For those that are above that threshold, you can deduct the taxes you pay in your home country agains what you would have been charged in the US. For almost everyone in the EU, that is a higher rate than the US taxes, so you don't pay anything on that European income.

_However_, you do still have to pay taxes on your US income if you're abroad. So if you are making money from freelancing with US companies (and, I assume, they aren't paying a European business you have set up), then you'll pay US taxes there, but due to tax treaties you are generally not double taxed.

You do, also, have to file every year.

0: https://www.taxfairnessabroad.org/blog/americans-abroad-by-t...