I am able to comprehend enough of your question to understand Safari is not loading websites because "Safari can't establish a secure connection to the server." Is that correct?
The reason is that the Safari version incorporated in OS X 10.8.5 is much too old to load recent websites. Nearly all websites use a security protocol that requires the use of a newer browser.
Your choices are to upgrade your Mac to a more recent operating system, or to find a browser that works with your existing OS X 10.8.5.
How to download and install macOS - Apple Support describes how to find a more recent macOS version compatible with your model Mac.
For alternative browsers newer than your version of Safari, consider Brave, or Firefox:
https://brave.com/
https://www.mozilla.org/en-US/firefox/new/