There are registry changes you could make but I couldn’t get those to work on my Windows 10 PC. To turn off private browsing in Firefox, you need to download a JSON file from GitHub.
You should no longer see the option to select Incognito mode in Chrome.
Repeat this from the Google key and name the new key Chrome ‘. This is fairly safe as long as you follow the instructions exactly. To turn off private browsing for Chrome, you need to make a registry change.