Sorry. Private Browsing can't be disabled. The most that can be done is to hide the menu items for it with the proper code, but the keyboard command would still turn it on and browsing history can still be cleared manually.

I'm not sure there is an official supported solution for this. What usually is recommended is to use separate parental monitoring software. E.g., Parental controls | How to | Firefox Help and Parental controls - MozillaZine Knowledge Base and PCMag - Parental Control & Monitoring - Products.

The following thread describes two other options: permanently disable private browsing | Firefox Support Forum | Firefox Help. The first (editing menus) is specific to the Firefox profile you edit. It would not affect other Firefox profiles or Safe Mode. The second (removing a program file) could be "repaired" by reinstalling the browser. So if someone is very determined, they can work around those changes.

You might like to look at this recent answer to the same question with entries by myself, The ed-meister and cor-el. The short answer is that turning off your operating system with a password and supervision seem to be the only answers.

https://support.mozilla.com/en-US/questions/827537

Children of moderate ability can install and uninstall a new browser in no time flat. And can certainly delete individual history items, cookies, cache entries. I've heard a router suggestion but I don't know if the router lists can be doctored. Also, as I noted, you can do anything with a USB stick these days.