So what do mean by closing that screen when you close Safari? Don't you simply press the home button to leave Safari and that gets you out of it just like that?
Also, are you always using Private mode for browsing?
You mean when you close the last tab in Safari? And when you click on "Done" you just get a new tab in that case? Neither of those really close Safari, as you basically do that when you just press the home button to exit.
I'm still not sure what the issue is. Is it that you just don't want to be in private mode? In which case you can just tap on "Private" on the bottom there to exit that mode.