I recently revived my old Jekyll project started with Jekyll 3.x (now using Jekyll 4.2 and I forgot a lot). The generated site is OK if browsed with jekyll serve command, but if later I manually enter _site folder and click index.html the site ignores css rules and links are broken.
This makes me nervous how to deploy the site. What I'm doing wrong?
Your site is served from a development web server when you run jekyll serve. I'm assuming by click index.html, you mean you are opening the file in your web browser - this won't work well with the absolute URLs Jekyll creates.
You can see the difference in the URL bar: one will say http:// and the other will be file://, a URL like /css/main.css will likely only work through the web server (i.e. jekyll serve).
Assuming you are deploying the site to a web server, and it works with jekyll serve, I'd guess it's fine. I can't be sure, but in any case, You should probably back up what's on the server currently though so you can restore it.