replace mysql_real_escape_string for security purposes - mysql

Question

I was using a WordPress Plugin That broke on update to 4.3. The error was
mysql_real_escape_string(): Access denied for user
I found out that the error was because the mysql_real_escape_string is called from within MySQL and needs to be logged in before execution. Solutions were to include a mysql_connect before the mysql_real_escape_string, which solved the problem. But it seems from many of the comments, the mysql_real_escape_string an this solution should not be used for various security reasons. But I am not sure how to change the code to PDO::quote etc as I am not really sure whats going on. The query I am trying to change is
function custom_permalinks_request($query) {
global $wpdb;
global $_CPRegisteredURL;
// First, search for a matching custom permalink, and if found, generate the corresponding
// original URL
$originalUrl = NULL;
// Get request URI, strip parameters and s
$url = parse_url(get_bloginfo('url'));
$url = isset($url['path']) ? $url['path'] : '';
$request = ltrim(substr($_SERVER['REQUEST_URI'], strlen($url)),'/');
$request = (($pos=strpos($request, '?')) ? substr($request, 0, $pos) : $request);
$request_noslash = preg_replace('#/+#','/', trim($request, '/'));
if ( !$request ) return $query;
$sql = "SELECT $wpdb->posts.ID, $wpdb->postmeta.meta_value, $wpdb->posts.post_type FROM $wpdb->posts ".
"LEFT JOIN $wpdb->postmeta ON ($wpdb->posts.ID = $wpdb->postmeta.post_id) WHERE ".
" meta_key = 'custom_permalink' AND ".
" meta_value != '' AND ".
" ( LOWER(meta_value) = LEFT(LOWER('".mysql_real_escape_string($request_noslash)."'), LENGTH(meta_value)) OR ".
" LOWER(meta_value) = LEFT(LOWER('".mysql_real_escape_string($request_noslash."/")."'), LENGTH(meta_value)) ) ".
"ORDER BY LENGTH(meta_value) DESC LIMIT 1";
$posts = $wpdb->get_results($sql);
return $query;
}
Is there an easy way to replace the mysql_real_escape_string or better way to do this? I am not sure this question is similar, but I dont see how to implement the answers.