May be you should consider rejecting random urls from your thumbnail generation endpoint. Now I could generate thumbnail of any website e.g. my blog https://hnews.xyz/thumbnail/?url=http://rajeeshcv.com
You could also tail the Firebase feed from a server process, generate the relevant images and only serve images for pre-existing urls.
You could try generating a unique token for each page-view, and use this as an access-token for all other resources pulled by the page.
The reality is that it's not too many people who will misuse your API... so don't pick a mechanism that will place undue load on your server until you really have to.