Googleling my website link doesn't show up

site: There is a link to the site shown, but a Google search doesn't show a link to the site. How is this going? Website penalized?

if it shows up for site: then it is indexed and most likely not penalized (especially if it's a fresh site - like few days / weeks old - I don't see how it can be penalized when google barely knows about it.)

Also, did you check the site in quotes? It would probably come up anyway even if you didn't put it in quotes (most of the time it does come up), but for extra safety you should check it with quotes around it.

And did you submit a sitemap to GSC? It's not mandatory, of course, but submitting the sitemap helps google index and figure your site out faster, so you probably want to do this if you're a newbie