Actually, they are superior in their own area. The Church (Pope) is superior in things spiritual and the State (King or Govt) is superior in things temporal.
This subject was explained quite successfully by St. Robert Bellarmine and Francisco Suarez in the early 1600s when they responded to James I's claim that he had a divine right to force the Catholics to swear allegiance to him because of his "spiritual" superiority.
The Church was given her command by Jesus Himself, whereas the king receives his authority from man and not God. Therefore, the Church is superior in the spiritual realm and the king has no divine right to rule.
Bellarmine made another very interesting point which was quite novel at the time: If the Pope is superior in things spiritual and the king is superior in things temporal, then the Pope has indirect spiritual authority over the king in things spiritual. So, each has primary authority but in different arenas.
For your consideration:
…However, one sword ought to be subordinated to the other and temporal authority, subjected to spiritual power. For since the Apostle said: 'There is no power except from God and the things that are, are ordained of God' [Rom 13:1-2], but they would not be ordained if one sword were not subordinated to the other and if the inferior one, as it were, were not led upwards by the other.
For, according to the Blessed Dionysius, it is a law of the divinity that the lowest things reach the highest place by intermediaries. Then, according to the order of the universe, all things are not led back to order equally and immediately, but the lowest by the intermediary, and the inferior by the superior. Hence we must recognize the more clearly that spiritual power surpasses in dignity and in nobility any temporal power whatever, as spiritual things surpass the temporal. This we see very clearly also by the payment, benediction, and consecration of the tithes, but the acceptance of power itself and by the government even of things. For with truth as our witness, it belongs to spiritual power to establish the terrestrial power and to pass judgement if it has not been good. Thus is accomplished the prophecy of Jeremias concerning the Church and the ecclesiastical power: 'Behold to-day I have placed you over nations, and over kingdoms' and the rest.
Boniface VIII, Unam Sanctam (One God, One Faith, One Spiritual Authority) November 18, 1302
This says close to the same thing you're asserting, but showing that there is a more direct link.
Let everyone be subject to the governing authorities, for there is no authority except that which God has established. The authorities that exist have been established by God.