Highly unlikely. A site could try to exploit unpatched security holes in your browser, but if your browser is up to date, this is unlikely to succeed. Modern browsers are very complex and large so they have lots of weaknesses, but they also get fixed quickly, a lot of eyes are on their code and they utilize sandboxing techniques as well to isolate things from your system.
Still, it’s a good idea to harden your browser further yourself, or run it in an additional sandbox.

Check Flatpaks as well.