See if this addon does what you want.
https://addons.mozilla.org/en-US/firefox/addon/disable-private-browsing/?src=ss

Any changes can be reversed by anyone with moderate skill.

Just remember that another browser(e.g. Chrome, Opera etc) can be installed and uninstalled in no time flat. I won't publicize what you can do with a USB stick.

If your problem is what I think it is, a password on your computer or supervision are the best solutions. It would also be worth your while to search through "parental controls".

There are a lot of ways to clear such data, either via Clear Recent History or directly by deleting the History in the Library or sidebar.
Private Browsing makes it easier because all data is kept in memory and nothing is saved to disk, so nothing needs to be removed.

If you want to see which sites have been visited then you should look at others means outside Firefox like a router or firewall with a password protected logging feature that can't be bypassed or a program like Wireshark to log internet access.

• Wireshark: http://www.wireshark.org/download.html
• Wireshark - Wikipedia: http://en.wikipedia.org/wiki/Wireshark

See also:

• http://kb.mozillazine.org/Parental_controls