Like many people in this subreddit I have been trying and failing to add my amex gift cards to my Amazon account, I did some digging in this sub and I found a solution buried in a comment thread somewhere, special thanks to u/jorgejdejesus for being the person to comment about the solution originally.

Step 1. Add the amex gift card to your amazon account just like you would do with a regular credit or debit card (Click account, then click Your Payments).

Step 2. Go to the Gift cards section under your account, and click on reload your balance. You will have to then enter a custom amount of $9.99, and then click on Buy Now. Once you have done this make sure you select the amex gift card as the payment option, and purchase the reload.

Repeat this step until the gift card is empty, if you don't want to do the math in your head every time you can just use this link and enter in your gift card information to see your remaining balance. Eventually you will only have a few cents left on the card but don't forget to add those too!

(Optional) Step 3. You don't always have to just keep adding in increments of $9.99 after your initial $9.99. The max amount I was able to put on at once for a $50 gift card was $39.91 and on a $100 it was $49.99.

I suspect that you could find higher increments but this method already takes some time to go through a single card so I didn't mess with it more to see what the limits were. Be warned I did lose a random $0.10 once after a higher amount I entered didn't go through but it's only $0.10 so whatever.

That's it! Now you can use your amex gift cards on Amazon.