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Keep in mind not all websites allow you to stay logged in without activity including Apple Support Communities.
If you are actively using a website and still get logged out, go to your Safari menu bar click Safari > Preferences then select the Privacy tab then click: Remove All Website Data.
Quit and relaunch Safari. If the same thing happens again, you may have a Safari extension or third party plug-in installed that's causing a conflict.
It's easy to troubleshoot. From the Safari menu bar click Safari > Preferences then select the Extensions tab.
Uninstall one extension at a time, quit and relaunch Safari to test. If it's not an extensions issue, try troubleshooting third party plug-ins.
Back to Safari > Preferences. This time select the Security tab. Deselect: Allow plug-ins. Quit and relaunch Safari to test. If that made a difference, instructions for troubleshooting plugins here.