Twitter supports uploading a photo when posting a Tweet. These photos may appear in a timeline and are affected by an “auto-expand photos” preference in your user timeline widget settings.
Twitter also supports link previews powered by Twitter Card markup. An individual Tweet link may display a Twitter Card with more details about a link contained inside the Tweet. That link preview may also be visible when embedding an individual Tweet on your website.
Tweets created by Facebook as a result of creating a new post on your Facebook page are the responsibility of Facebook. If you believe Facebook previously did a better job posting your Facebook Page updates to your Twitter timeline you should contact Facebook support to communicate your preferences.
Example of Tweet with a photo upload by NYTimes:
Example of a Tweet with a Twitter Card displayed for the same shared nytimes.com article:
https://twitter.com/myoldjeans/status/600004607163629571