If you are using the same Amazon account for all three devices, 1) computer, 2) iPad and 3) Iphone, all the books should sync across all three devices. Does this not happen?
On your iPhone, down at the bottom of the screen when you open the kindle app there is a button called “More.” Click on that and about half way down there is a button “sync.” Try that.
Here is a toll free number for Kindle: 866-321-8851. Amazon is pretty helpful over the phone. I’ve called them many times when I have problems. They could walk you through anything like this.
Good luck.
FYI: there is another app called Libby. It will allow you to connect to your library through your library card. They will have a gigantic collection of books and audio books you can download for free. We use that instead of Kindle. Your library should know about this if you have any questions. But your library card number is your log-in for Libby.