Sanırım çalışır:
SELECT u.id, u.Yonetici,
COUNT(h1.*) AS ekledigi_haber,
COUNT(h2.*) AS onayladigi_haber,
COUNT(y.*) AS sildigi_yorum
FROM user AS u
LEFT JOIN haber AS h1 ON (u.id = h1.insert_user_id)
LEFT JOIN haber AS h2 ON (u.id = h2.update_user_id)
LEFT JOIN yorum AS y ON (u.username = y.update_user)