Stay safe - avoid completely disabling your antivirus applications and/or the web shield features. Make the smallest changes first, then turn things off only if absolutely necessary.
Instead, check your AV app for a place/way to specify safe sites. In my case, I could not get to a banking site. I suspect the problem is that they are using SSL, not TLS for securing their server, but regardless, here is the step I took that fixed my problem (had the same error about Safari can't establish a secure connection):
- Open the AV app
- Look for settings or preferences (Avast: Preferences)
- Locate the web configuration section (Avast: Web Shield)
- Locate the configuration change feature (Avast: Settings)
- Locate the add trusted site feature (Avast: "+" symbol)
- Type the server name; for example, "www.news.com", or "www.appleheadaches.com", or "geewhiz.org".
- Select whether it's a regular http site or a secure https site. Hint: if it's causing headaches, it's probably an https site.
- Save the new site (Avast: click the Add button).
- Profit!!!
That fixed my problem in seconds. I no longer get the Safari connection error; I get to the site I was trying to load in the first place.
And my security controls are still in place, keeping my *** out of trouble!
Hope this helps!