I want to make the user profiles private. In such a way that if someone clicks on your username, they cannot see the publications that user has made, or anything.

Is there any extension? Is it possible?

    GreXXL What a shame, I just installed it but it doesn't do anything. It doesn't seem to work.

      Code1990 as I can remember Flarum also have a separate permission for viewing Profile Pages. Maybe it is all you need (hide from some parts of users: unregistered, registered, some groups, etc).

        Gorm Flarum also have a separate permission for viewing Profile Pages

        No, there's only a built-in option for searching users. In older Flarum versions it was named in a very confusing manner that suggested it might control profile visibility but it was never the case.

        It shouldn't be very difficult for an extension to achieve this. Removing the profile page might be troublesome, but preventing to see the discussions and posts of a user should be simple, it's just a search gambit that could be disabled depending on the permissions, and some visual tweaks to the profile page to say that you are not authorized to see the content instead of showing "0 discussions".