In 2010, you came across a site that was giving away bitcoins. The site said it was free money so you signed up.
A balance on a site is just virtual until it sits in your wallet. Not your keys, not your wallet.
How would you recover your BTC?
There might be no way to locate such website again because most websites that gave free bitcoin during that time are gone.
Just accept your fate and move on because the biggest mistake you made was leaving it on website.