That’s not how a website works. For a server hosting a web page with an embedded stream it only does minimal work two times. Once when the page layout is requested and the other when the API request returns. The actual audiovisual content never pass through your server, it is only between Twitch servers and your visitors. It is only when/if your server receives many page load requests from many visitors at the same time that it becomes stressed.
EDIT: with javascript/jquery the clients browser will handle the entire api request client side so that means even less work for your server. It will only provide a few written documents of html + js at a few kilobytes.