You need to use a direct link to an image file, which usually ends in an extension like .png or .jpg. You may need to jump through hoops to get it on sites like Pinterest where a right-click and "Copy image link" doesn't work (inspecting the element or network transfers usually does). Some image hosts (notably i.redd.it since 2023) don't allow hotlinking/embedding from outside domains as per their CORS policy, in which case you get a result like this:
It is impossible for a Lemmy web client to fix this without a browser extension but apps like Imgur Viewer can (and do) spoof the content-type and referrer HTTP request fields.
Some clients (notably Boost) will automatically make any link into an embedded image if it contains a known image file extension, which is unhelpful if the link leads to a HTML page instead (such as . Others refuse to load an image if its URL does not end in one of the known extensions (if the post URL doesn't, a link post will have a "↗⃞” external link icon instead of an image) but you can usually trick it by appending ?. png or similar to the URL.
Interesting. On my client (Eternity) both the picture from Reddit and the picture from gify.nou.cz load. On the Lemmy website only the Reddit image loads.