How do I clear cookies from the Twitter app internal browser on iPhone?

I don't believe there is anything you can do sans uninstalling and reinstalling the Twitter app. Each application that uses the WebView has its own cookie store.

A developer can programmatically remove cookies. So the Twitter app would need to give you a way to clear cookies which I don't think it does.

In the Twitter app, open Settings and privacy. As of August 2017 and version 7.4, it is accessed by tapping your profile picture in the top-left corner.

Now go into Data usage → Web storage and select Clear all web storage. This will delete your Twitter cache, cookies and logins.

That action does not affect pictures, videos and vines attached to Tweets. Instead those are deleted under Data usage → Web storage → Clear media storage.