Using count in triggers to update another table
This trigger does the job:

create trigger trx_updates_atrig
after insert on trx_updates for each row begin
DECLARE updatecount INT;
  set updatecount = ( select count(*) from trx_updates where request = new.request );
  if updatecount=4
    then
      update trx_request set finished=NOW() where id = new.request;
  end if;
end//


SQLFiddle Demo.