Use array notation like name="checkbox[]"
in your input
element. This will give you $_POST['checkbox']
as array. In the query you can utilize it as
$sql = "DELETE FROM links WHERE link_id in ";
$sql.= "('".implode("','",array_values($_POST['checkbox']))."')";
Thats one single query to delete them all.
Note: You need to escape the values passed in $_POST['checkbox']
with mysql_real_escape_string
or similar to prevent SQL Injection.