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.
here i have interesting short story , i was so dumb in 2013 bought 10 bitcoin for 60 usd each .
found a site that basically it was ponzi offering me to double my bitcoin in no time by sending to certain address , try it out for 1 bitcoin and successfully make it 2 in the next day , really amazed and repeat it several times but guess i successfully withdrawn 2 bitcoin while actually i have sent more than 5 bitcoin and get held there until now lol i have tried to tracing back the site in 2015 and of course they are disappeared haha
the good old dark days

. it was really fun experience, in that cases you should not take it seriously. there is a lot of free bitcoin events in the past and they are already gone , pretty simple to solve it, forget everything.