Firing up an old Mac G3, set up the same as Jack Campin (MacOS X10.4.11 and Safari 4.1.3) , I get the same result. Can't establish a secure connection.

But no problem with Safari 10 on my iMac running OS X 10.10.5.

I get the identical result with my own website, www.johnroberts.name. That's hosted by Karelia, which is a simple link from Sandvox where I created the page. (A simple web design program for simple minds).

Could it be old technology in Safari (e.g. I run an old version of Adobe Acrobat that tries to check for updates every week, and tells me I have no internet connection).

Safari was the only browser loaded on my G3. I tried to download Firefox, but got the message that my system didn't meet its requirements. I did manage to download Chrome, but got a similar message when I tried to open it.