I don't think there are any certain way to disable this private browsing mode. But I found an addon that will disable this mode and will force user only to surf internet in standard mode. You may try this. I have tested this and it works well :-)

This addon removes private browsing option from the menu and also disables its keyboard shortcut (CTRL+SHIFT+P).

Download it from here: https://addons.mozilla.org/en-us/firefox/addon/disable-private-browsing-pl/

WARNING: YOU WILL NOT BE ABLE TO UNINSTALL THIS ADDON UNLESS YOU RESET THE BROWSER OR FORMAT THE SYSTEM.

Hello, see also : CCK2 Wizard extension

Note also that EVERY extension is disabled (and....useless) in firefox safe mode.

thank you

Note that even if you would disable Private Browsing mode that it is still relatively easy to delete history manually in a lot of ways via Clear Recent History or the Forget toolbar button or in the History Manager.

You would need a router with a password protected logging feature to have a log of all visited website.