How to order custom fields in the user profile (admin area)

Unfortunately there’s no action hooks on user-edit.php that would allow you to place html before the profile image. One option is to use javascript/jquery to move the html around the profile page.

But as @RiddleMeThis mentioned on the comments, you can use user_contactmethods filter to add custom contact methods such as Twitter to the profile page.

add_filter('user_contactmethods', 'my_user_contactmethods');

function my_user_contactmethods($contactmethods){

  $contactmethods['twitter'] = __('Twitter Username', 'textdomain');

  return $contactmethods;
}