The only way to really try to prevent your kids from browsing areas of the internet that you don't want them to to is to install a parental control content filter. Even if there were a way to disable private browsing, it could be easily worked around.

I'd suggest you research parental control software and use the one that best fits your needs.

Also, please note that your Firefox version is out of date. Please Update Firefox to the latest release

This does't answer my question. Can private browsing be disabled?

I said above:

Even if there were a way to disable private browsing, it could be easily worked around.

There is no way to turn off private browsing that could be worked around with a new profile, or using another browser (all modern browsers have private browsing and it cannot be disabled in any of them). That's why I suggested a content filter as a practical solution to your problem.

Even if PB mode would be disabled then they can still clear history manually in a lot of ways via Clear Recent history or via the Forget toolbar button or by clearing history in the history sidebar and in the History Manager and other data like cache and cookies via their respective buttons in the user interface. You can only monitor what websites they visit via software that runs on the OS level.