If I do a post on Twitter and include a link, a preview of this linked page shows as a preview image on the post. On Facebook you can just remove this using the dropdown menu of the post - I can't see how to do this on Twitter thought. Is it not possible?
It doesn't appear as though you can do it from the web app. However, you can do it from the iOS and Android Twitter apps.
In each of those mobile apps, when you paste a link into a draft tweet, there will be an X in a black circle in the upper right corner of the preview. Click the X to remove the preview (but keep the link).