Is this a correct use for
Many websites could have gone down during the years.
What I want to say, is the websites do not exist any more. I want to use
gone down, but not sure if it is correct here.
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Gone down would mean that the people maintaining the website were no longer able to keep maintaining it and it eventually went offline. This is more frequently used for saying that a website has gone down due to technical issues or even hacking.
Google has gone down due to technical issues in the servers.
You could've also said that "many websites have been taken down over the years", that would mean that maybe the government or some higher power took these websites from the web because of criminal activity or misuse of information.
MegaUpload has been taken down by the government after allegedly uploading illegal content to their servers.
It all depends on your context, but there, are two expressions you could use.