On the form, change
action="<?php echo site_url("https://wordpress.stackexchange.com/"); ?>"
To:
action="<?php echo esc_url( get_post_type_archive_link( 'club_societies' ) ); ?>"
Where 'club_societies' is the slug of the post type you’re searching.
That will set the form action to the archive URL, with the search parameters added, rather than the homepage with search parameters added.