Worried your spouse is cheating on you?
Disable Inprivate browsing:
http://www.thewindowsclub.com/disabl...chrome-firefox
Disable History Cookie Deletion:
https://technet.microsoft.com/en-us/.../dd349812.aspx
If you don’t want users to be able to delete their browsing history, form data, or passwords, you can enable the following group policy settings located in both Computer Configuration\Administrative Templates\Windows Components\Internet Explorer\ and User Configuration\Administrative Templates\Windows Components\Internet Explorer\:
Turn off “Delete Browsing History” Functionality
Turn off “Delete Forms” Functionality
Turn off “Delete Passwords” Functionality
Did you check (in IE11) Tools/Internet Options/browsing History? There is a checkbox for deleting browsing history on exit.
Charlie, you can add GPEDIT.MSC as I mentioned above. I tried it on Win 10 Home and it does work. After you have it, you can try what PCGEEK11 suggested
Turn off Delete Browsing History Functionality
Turn off Delete Forms Functionality
Turn off Delete Passwords Functionality
Computer Configuration User Configuration
-> Administrative Templates -> Administrative Templates
-> Windows Components -> Windows Components
-> Internet Explorer -> Internet Explorer
-> Delete Browsing History -> Delete Browsing History
From post by pcgeek11
If you don’t want users to be able to delete their browsing history, form data, or passwords, you can enable the following group policy settings located in both Computer Configuration\Administrative Templates\Windows Components\Internet Explorer\ and User Configuration\Administrative Templates\Windows Components\Internet Explorer\:
Turn off “Delete Browsing History” Functionality
Turn off “Delete Forms” Functionality
Turn off “Delete Passwords” Functionality

I've tried a few regedit DWORDs and I haven't been able to disable it completely... anyone?
Also... is there a way to disable deleting the browsing history?
It's not private anyway. Even when you browse privately, it still stores the sites you visited. There is a thread about this somewhere.
Fixed issue with Microsoft Edge browser caching visited URLs while using InPrivate browsing.