I have an old iPad, so I'm not sure if this is a problem with legacy hardware, but I'm feeling very incompetent and frustrated right now. I've already tried everything discussed here before seeing this post. If I click the private button, nothing happens except the banner goes black at the top where you enter your search. If I go to the Favorites Menu, I see all the bookmarks but nothing to edit the list. I've tried holding an Icon until it expands, but there isn't an x button to delete it. I've tried swiping the icon off the screen, and this does nothing. I've gone into the settings and selected Safari, then favorites, but there is nothing there but a list of all kinds of favorite folders from who knows where or when. Several Favorite folders with Popular, Apple, News and Popular. I've noticed the favorites change depending on which of these folders I select. I'm guessing they are from syncing with Google Chrome and all my other browsers across all of the devices I use in the iCloud to sync.
I'm mystified how anyone gets from their final a ha I've figured it out to actually deleting anything. SO, I've gone to my iMac and deleted all the favorites I don't want in Safari. But that has yet to sync with the iPad and remove the confusing list of favorites in Safari Settings, Favorites.
So this problem is not solved for me.