Twitter's implementation just exploded on me the other day on my phone. Since then I cannot open ANY links to twitter anymore. I just get a page from the browser that says "Cannot complete request".
It's on my phone so I can't just clear a single site's data, and I cannot be bothered to clear everything and relog everywhere.
Hook up your iPhone to your Mac via lightning cable. Open Twitter on your iPhone and then on your Mac open Safari, enable develop mode, and then go to Develop > Your iPhone > Twitter tab. This will open up inspector on your Mac allowing you access to the local Twitter data on your iPhone.
It's on my phone so I can't just clear a single site's data, and I cannot be bothered to clear everything and relog everywhere.