Can posts show branded link instead of just photo with short link?

My posts come out and there’s the image and the link is so small. No one is engaging with these posts but when I post the link directly to facebook and there is a whole LINK with a frame and title and it looks better, people engage. How do I get the real deal that shows more of my site name and meta description when sharing?

This is what I want.

THIS IS WHAT I GET (What I do not want)

I would like the content to have my brand name on it. Is this possible?

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