When you add an attachment to a feature, it gets displayed in the pop-up window as a link that allows you to download a file (in your case, an image). Attachments can be other file types to, so arcgis.com map viewer can't draw them in the pop-up. To see the download links inside the pop-up window, you need to turn on attachments when you configure popups, see attached image.
Pop-ups also support pop-up media (images and charts) that do get displayed in the pop-up. For images, they need to be web accessible images that you reference via a URL.
Hope this helps,
Mike