You can't do anything when browser closing, because for doing something, some PHP script must be executed. Other words you can't doing anything till you call some url on your website.
Did you try set expire_on_close
to true in config/session.php
?
/*
|--------------------------------------------------------------------------
| Session Lifetime
|--------------------------------------------------------------------------
|
| Here you may specify the number of minutes that you wish the session
| to be allowed to remain idle before it expires. If you want them
| to immediately expire on the browser closing, set that option.
|
*/
'lifetime' => 120,
'expire_on_close' => true,