I can't create folders or add bookmarks on my iPhone with the book icon

Hi Jemmaker, you should be able to add bookmarks using the typical process described in Bookmark a website in Safari on iPhone - Apple Support:

1. Tap the Book icon.
2. Tap Edit, then do any of the following:

• Create a new folder: Tap New Folder at the bottom left, enter a name, then tap Done.
• Move a bookmark into a folder: Tap the bookmark, tap below Location, then tap a folder. Tap ( < ) to return to your bookmarks.
• Delete bookmarks: Tap the Minus icon, then tap Delete.
• Rename bookmarks: Tap the bookmark, enter a new name, then tap Done.
• Reorder bookmarks: Touch and hold Sandwich icon, then drag the bookmark to a new location.

I just tried using these just now on my iPhone and they seem to be working. If these options don’t work for you, let us know what step of the process doesn’t work and we can troubleshoot some more.