Can you share links to a few of the websites? We can check them to see if we get the same issue.

It might be that the security certificate for those sites has expired. When that happens you get a security error like you've seen warning you the site may not be safe to visit. But in the cases where the site has forgotten to update their security certificate, the site should still be safe. Unless it is a banking site or anything involving money, I would not login to that site until they have fixed their security certificate. You could be at risk if you login to a site while their certificate is not updated.

Suddenly, the problem disappeared for all of my web sites. I have no idea why. But, the problem is "over" even if not solved.

Wow! Good to know. If it returns, please let us know. Also if you have any Antivirus software installed, sometimes they can cause these security errors: https://support.mozilla.org/en-US/kb/error-codes-secure-websites