Output user_url without http://

I’m assuming WordPress is applying esc_url() to that value to ensure a valid URL. A domain would not qualify as a proper URL and therefore esc_url() adds http:// to it to ensure it’s a URL.

See https://codex.wordpress.org/Data_Validation for more on why they force http://.

There are a couple things you can do.

One is replace http:// in the string with nothing on the front end:

$replacements = array(
	'http://' => '',
	'https://' => '',
);

echo strtr( $user_url, $replacements );

That, of course, would honor WordPress’s addition to http:// in the database, but replace it on the front-end.

If you want the DB stored value to not have http:// you’ll have to do something like this:

function dont_http( $raw_url ) {
	$replacements = array(
		'http://' => '',
		'https://' => '',
	);

	return strtr( $raw_url, $replacements );
}

add_filter( 'pre_user_url', 'dont_http' );

See https://developer.wordpress.org/reference/hooks/pre_user_url/ for more.

The other option is not use $user->user_url. You could create a new User Meta field called user_domain where the http:// won’t be added automatically, it will be whatever you store in there.

http://wpengineer.com/2173/custom-fields-wordpress-user-profile/ has a pretty simple guide on adding custom meta for users.