For the private browsing,
Go to Safari and select the multiple tabs icon bottom right (page behind a page).
You should then have three options across the bottom, select Private on the left.
Tap this and read what it says. It should have "Close All Pages?" In bold and the "would you like to close your existing pages before..."
If it says "...turning off Private Browsing" turn it off and see if Safari remembers now.
However, if it says "...turning on Private Browsing", it's should still be an easy fix in the settings, you'll just have to set a passcode.
So if Private Browsing isn't active, go back to Safari>Passwords and Autofill
Just underneath Names and Passwords is "Always Allow". However, to enable that you have to set up a passcode. Set one up and see if that works, if that doesn't work give us a shout back.