It's actually possible to have infected media file (such as pictures). In the case of pictures, it's done via what we call "stenography" and the malicious code can leverage a known vulnerability in the application used to handle that picture file, and from there, execute malicious code that can lead to infection.
What told you exactly that the picture transferred you to an unknown site which tried to download suspicious files? And like I said, I don't think it's possible since unless the leverage an exploit on Twitter, Facebook, etc. and these have yet to be found (in the context you're referring to). Taking screenshots of a picture (using a screenshot software like ShareX or the Print Screen button) is safe to me since no code gets executed, except the one of your screenshot software.
Like I said before, I've never heard of a such case, so I'll wait for an actual employee in the Malware Research team to answer that one.