Can NOT search my custom post type?

did you try if (!isset($post_type)) or if(empty($post_type))?? For example I would try this:

    function mySearchFilter($query) {
    $post_type = $_GET['post_type'];

    if ($query->is_search) {
        if (!empty($post_type)) {
           $query->set('post_type', $post_type);
    }
  }
    return $query;
}

add_filter('pre_get_posts','mySearchFilter');

EDIT: Here is my working example code:

Here is my CPT used for testing:

add_action( 'init', 'register_cpt_website_bookmarks' );
function register_cpt_website_bookmarks() {
$labels = array(
'name' => _x( 'Bookmarks', 'website_bookmarks' ),
'singular_name' => _x( 'Bookmark', 'website_bookmarks' ),
'add_new' => _x( 'Add New', 'website_bookmarks' ),
'add_new_item' => _x( 'Add New Bookmark', 'website_bookmarks' ),
'edit_item' => _x( 'Edit Bookmark', 'website_bookmarks' ),
'new_item' => _x( 'New Bookmark', 'website_bookmarks' ),
'view_item' => _x( 'View Bookmark', 'website_bookmarks' ),
'search_items' => _x( 'Search Bookmarks', 'website_bookmarks' ),
'not_found' => _x( 'No bookmarks found', 'website_bookmarks' ),
'not_found_in_trash' => _x( 'No bookmarks found in Trash', 'website_bookmarks' ),
'parent_item_colon' => _x( 'Parent Bookmark:', 'website_bookmarks' ),
'menu_name' => _x( 'Bookmarks', 'website_bookmarks' ),
);
$args = array(
'labels' => $labels,
'hierarchical' => false,
'supports' => array( 'title', 'editor' ),
'public' => true,
'show_ui' => true,
'show_in_menu' => true,
'show_in_nav_menus' => false,
'publicly_queryable' => true,
'exclude_from_search' => false,
'has_archive' => true,
'query_var' => true,
'can_export' => true,
'rewrite' => true,
'capability_type' => 'post'
);
register_post_type( 'website_bookmarks', $args );
} 

Link: [link removed] (search for “Bookmark” or “Test” and if you remove the “post_type” and search for test you’ll get more results)