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I deleted my old site entirely and made a new site 8 months ago.
But still in the crawl errors section of Google Search Console, a lot of 404 and not found errors appear. All the URLs are for the old site.
How long does it take for Google to stop giving these errors? What can I do in the meantime?
how long does it take for google to stop giving these errors?
There is really no fixed time. For some of the more 'active' websites, I have noticed Google to make alterations to search index at a fast pace as compared to less visited (by Google bot). As correctly pointed out in comment section, wait it out.
what can i do meanwhile?
If all the urls being shown as 404 are indeed not available then skip the following suggestion. If not, try an redirect old url to new ones, this will help you keep traffic. Additionally, you can keep a check on indexed links and see if Google is removing them (or not).
P.S - If you have a relatively young website, this can take some time.
Once Google crawl your webpages and index it on their database, then they will continuously visit your webpages to check weather you have made any changes or not. If you made any changes like changing article paragraph, then they will update that changes on their database.
Now when you remove that webpages, then Google think might be that page will be come back again in future and report you 404 error page in search console, so you know about that. (Generally it's not harmful, you might already read that line). So Here Google don't remove your webpages from their index database, instead they will wait for a long time, may be years, if your site domain information is not changed and you're still owner of that website. And you're domain name is not expired.
If your site domain is expire then Google will know that, yes now that webpage will never come back again, so let's delete it from our database, otherwise they will keep that webpages on their database, even when your site is down for many months, so when your site is down for many months, then Google show it, it's cached version of your website in SERP.
The simple solution is use 410 Gone error, that's error tell Googlebot to, those webpages are gone forever so don't report me 404 error, because they will never come back again.
Use 301 redirection only, if your webpages are similar, otherwise don't use it.
