For a mac air 2013. Safari not working

Have you checked to see whether the version of Safari you are using is listed as compatible with the website? If not, then you may need to update to a newer version of macOS so you can use a newer version of Safari.

Have you tried deleting the Cookies for the website and deleting the browser cache? Make sure to close the tab for the website first.

You may need to use a third party browser such as FireFox, Vivaldi, or Google Chrome.

Safari can’t open a website

 Safari – slow page loads

 Safari - Unsupported third-party add-ons may cause Safari to unexpectedly quit or have performance issues

Safari doesn't load a page or webpage items are missing

Web Content not responding

Try setting up another admin user account in System Preferences/Users & Groups to see if the same problem continues. Please post back on whether or not this worked. Also try the Safe Mode. Please post back on whether or not this worked.

Isolating an issue by using another user account 

Safe Mode - About

If it works in the Safe Mode, try running this program when booted normally and then copy and paste the output in a reply. The program was created by Etresoft, a frequent contributor.  Please use copy and paste as screen shots can be hard to read. Click “Share Report” button in the toolbar, select “Copy to Clipboard” and then paste into a reply. This will show what is running on your computer. No personal information is shown. If the log won’t post, try posting it in Pastebin and provide a link in a reply. After pasting the report in a PasteBin page, go to the top of the page, and copy the address in the URL bar. Paste that in a new reply.        Pastebin

Etrecheck – System Information