I understand that Mac is running Lion, but you quickly determined its version of Safari is practically useless. You are unable to upgrade it by following the instructions Apple provided in How to download and install macOS - Apple Support because Safari "... can't establish a secure connection to the server support.apple.com" or words to that effect.
Is that correct?
Please open the Terminal app - it is in your Mac's Utilities folder, and looks like this:

- To open the Utilities folder, select the Finder's Go menu > Utilities.
Select the entire line that follows below (triple-click to select the entire line), select Edit > Copy, and then Paste it into the Terminal window:
curl http://updates-http.cdn-apple.com/2019/cert/061-41424-20191024-218af9ec-cf50-4516-9011-228c78eda3d2/InstallMacOSX.dmg --output ~/Desktop/InstallMacOSX.dmg
... then press the Return key.
Keep that Terminal window open. If some kind of error appears please let me know what it says (you can copy from Terminal and paste it here).
If all goes well a InstallMacOSX.dmg file will appear on that Mac's Desktop. Don't do anything with it yet. Terminal will show the file download progress with labels Time Total / Time Spent / Time Left which you can watch if you have nothing better to do.
Wait for the Terminal app to stop updating those values. It will eventually finish and show the prompt you saw when you opened it. When it does, double-click the InstallMacOSX.dmg file to mount that disk image. Double-click the package installer within it. That will initiate an upgrade to OS X 10.11.x "El Capitan" which will be required prior to upgrading to anything later.
- El Capitan will incorporate a (slightly) more usable version of Safari.
Please let me know if that works. If it does, then you will be able to proceed with upgrading to High Sierra through the App Store in the usual manner.