On iOS, you can often tap a user's profile photo within the app to display it "all over the screen," making it easier to view or screenshot.
If you open Threads in a mobile browser (rather than the app), long-pressing the profile picture may trigger a "Download Image" or "Open in New Tab" option that the app blocks. Image Quality and Formats threads profile picture downloader full
Small circular icons often hide the details of complex logos or artistic portraits. On iOS, you can often tap a user's