The problem is not a safari problem or anything complicated like DNS etc.
When you close an APP the IOS saves a screen shot of the state of the app on closure (this also happens if the iPad / phone is locked or autolocked). When you return to the app the the IOS shows you the saved screen shot whilst it re-activates the app and the sends you the current screen. The big problem occurs when you change oriantation (ie from landscape to portrait and even when you rotate the device 180degrees) The IOS remembers the screen shot from each oriantation of the screen for example:
You are using your ipad in private mode with the screen in landscape, you then change to another app and move the screen to portrait for the 2nd app (the IOS has now taken a screen shot of your ipad's private mode in landscape view)
You then stay in portrait mode, switch back to your private browsing, finish what you were doing and then quit private mode thinking that you are secure...
When you next open the app in landscape mode it will call up the saved screenshot of landscape (which was your private browsing) before then updating the screen hence the problem
I have found that opening the app, rotating the screen 90degs closing it and then repeating that operation for the 4 screen orientations then updates the saved image of each orientation deleting what was previously buffered
This happens to every app and there is nothing that you can do to stop it, it will require apple to mod the IOS to not remember screenshots of apps in private mode. The whole reason for this is to give the user the impression that their ipad is super fast as the app opens instantly but what is really happening is your being shown a "last seen" screenshot of the app whilst it loads in the background
I dont know how many people have to complain before apple do something?