How do I disable private browsing in a way that my kids can't just turn back on?

This is complicated. Just disabling that option doesn't mean there's not other ways to flip the prefs to "never remember history" basically resulting in the same, or, when in normal browsing to be able to redact history and site data, either by domain, or by time spans — or by just pressing the "Delete" key when in the browsing history library pane reachable from the main menu. And even if you hide all the buttons and disable their key shortcuts (that, well, can be enabled back for the session by anyone, and then removed to leave no trace, if anyone's so inclined), there's still the possibility of sanitizing the browsing data using e.g. an addon… so I'm not sure the ask to just disable private mode is the solution for you — you might need to look more into enterprise policies to see what all can be disabled/removed from the installation, but I'm afraid you'd be playing whackamole — and if the kids see that as a challenge to try to circumvent the restrictions… well, I'm afraid you'd be the one playing catch up;)

Not sure I have good advice for this, the above is merely a reminder that removing specific parts of the functionality does not eliminate the whole of the privacy options available in the browsers.

Choice a 'family-safe DNS' and for more control 'family-safe firewall' , you may need to setup your own Ingress/Egress filters. In Firefox; Menu>Settings>DNS over HTTPS>Max Protection>Choose Provider>Custom>{add your chosen family-safe DNS DoH service url string here}

Thank you both. Bigger than disabling specific features in Firefox is the question of how best to curate as safe as possible an internet browsing experience for my family. I understand this is a broader question than twiddling the knobs in FF. I have controls set at the router level, but they are clunky and inconsistent. It would be really great if someone would make a simple to use internet portal that had our flourishing in mind rather than profit. FF still feels like a step in the right direction, even if it's still not exactly what I want.

This is complicated. Since the browser is very privacy–oriented, it actually goes a long way in the end–user's ability to conceal their traffic from prying eyes, be that a service provider, employer, school administrator… and unfortunately in your case the same applies to a parent or anyone else in the household in general.

To some extent you might be actually looking into a "kiosk mode" or a guest mode if you will, so that the user when unsupervised can't change the browsing settings arbitrarily — which could circumvent any protections you might wanted to have in place.

Unless you'd want to analyze traffic logs on your router, I'd actually perhaps advise to look into the capabilities of Enterprise Policies, which can "lock in" some of the prefs in a way "unprivileged" end user can't change, and look into 1.) all things pertaining private, sanitization or clear on shutdown features, and potentially disabling any data removal (this bit I'm quite unsure about) — 2.) setting some family–friendly DNS–over–HTTPS filtering servers and also locking these in place.

That said, I'm not sure if you'd be able to enforce all of that into one browser, and yet restrict other family members from either using a different browser, or even downloading say a "portable" copy of one and accessing any content they want, unsupervised. That all kinda depends on your "threat model";)

Bonjour

DisablePrivateBrowsing https://firefox-admin-docs.mozilla.org/reference/policies/disableprivatebrowsing/

HKEY_CURRENT_USER\SOFTWARE\Policies\Mozilla\Firefox\DisablePrivateBrowsing Or HKEY_LOCAL_MACHINE\SOFTWARE\Policies\Mozilla\Firefox\DisablePrivateBrowsing

Your browser is being managed by your organization

Nathan, you might search of: 'family-safe dns & firewall appliance'