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;
}