The tweets embeded in threads are in iframes. The iframe source is twitter's servers. They format the tweet and render into the iframe so it shows here.
When they see a link in the tweet, like
www.sportstwo.com, their server fetches the html at that URL and saves it on their hard disk. They show that HTML in the tweet iframe.
Since the tapatalk plugin was basically first thing in the HTML, that's what twitter shows in the iframe - the ad.
The tapatalk plugin is gone from the HTML. I showed you that a few times here.
The twitter server fetched the HTML an hour ago before I took out the tapatalk plugin. That HTML has the plugin ad in it. So that's why we still see ads for URLs you posted already.
The reason they fetch the HTML and store it on the hard dive is that if there are 50 people viewing this thread, there are 50 iframes (one in each person's browser). Rather than fetch the HTML from the link in the tweet 50 times (a load on the S2 server), they share the one time fetched version 50 times.