Hi: I tried everything in settings too and nothing worked there. I have an ipad retina and downloaded 7.0, everything was fine and then they had an update to it and that was when I had this problem. I read the other posts and decided to try the one and it worked. I will try to give you specific directions: hope it works,
Go to Safari
Clear history - page should be clear, nothing on it but tabs at top
Click on plus sign at top
Private should come in bottom left hand corner (on mine it just stays there)
Click on it
Click - keep all tabs open
Then you should be able to keep all passwords
I sure hope this works for you as it can be so frustrating. I worked at it for hours until I followed the other posters directions.
Good luck Rileydj..