There is an important sense in which SCG can only be understood as a single extended argument: that is, St. Thomas builds from what he has already said in drawing new conclusions, and he continues this course throughout the length of the book. Ultimately, you don't get to his final conclusions in book IV apart from what he said at the outset in book I. The upshot of this is that in some respects it can be difficult to extract a portion of his argument for separate consideration: what he is about to demonstrate is dependent upon what has already been said.
So it is, to a fair extent, with what St. Thomas has to say about the episcopacy and the papacy. It depends on what he has already shown in previous sections on the sacrament of Orders. It's somewhat difficult to just jump in. But I think what he has to say about the papacy in IV, 76 is sufficiently interesting that it's worth making the effort. By way of laying the groundwork, then, one can look more closely at what Aquinas says beginning with chapter 55 (on the suitability of the Incarnation) and chapters 56-73 (on the sacraments other than Orders), and arrive at 74, where he says:
It is, of course, clear from what has been said that in all the sacraments dealt with a spiritual grace is conferred in a mystery of visible things. But every action ought to be proportioned to its agent. Therefore, the sacraments mentioned must be dispensed by visible men who have spiritual power. For angels are not competent to dispense the sacraments; this belongs to men clothed in visible flesh [section 1].In other words, just as the other sacraments confer grace through visible things, they must be dispensed by visible men (rather than invisibly, as by an angel or something). But this requires that these men be equipped and ordered to do so. And since Christ himself would not be on earth to fulfill this task,
it was necessary that Christ should establish other ministers in His place who would dispense the sacraments to the faithful; in the Apostle’s words: “Let a man so account of us as ministers of Christ and dispensers of the mysteries of God” (1 Cor. 4:1). And so He committed the consecration of His body and blood to the disciples, saying: “Do this in commemoration of Me” (Luke 2:19); the same received the power of forgiving sins, in the words of John (20:2.3): “Whose sins you shall forgive, they are forgiven them”; the same also were given the duty of teaching and baptizing, when He said: “Going, therefore, teach ye all nations, baptizing them” (Mat. 28:19). But a minister is compared to his lord as an instrument to its principal agent, for, as an instrument is moved by the agent for making something, so the minister is moved by his lord’s command to accomplish something. Of course, the instrument must be proportionate to the agent. Hence, the ministers of Christ must be in conformity with Him. But Christ, as the Lord, by His very own authority and power wrought our salvation, in that He was God and man: so far as He was man, in order to suffer for our redemption; and, so far as He was God, to make His suffering salutary for us. Therefore, the ministers of Christ must not only be men, but must participate somehow in His divinity through some spiritual power, for an instrument shares in the power of its principal agent. Now, it is this power that the Apostle calls “the power which the Lord bath given me unto edification and not unto destruction” (2 Cor. 13:10) [chapter 74, section 2].And it was not enough for the apostles to be granted this authority:
One must not say, of course, that power of this sort was given by Christ to His disciples in such a way as not to flow on through them to others; it was given “for building up the Church,” in the Apostle’s phrase. So long, then, must this power be perpetuated as it is necessary to build up the Church. But this is necessary from the death of the disciples of Christ to the very end of the world. Therefore, the spiritual power was given to the disciples of Christ so as to pass on from them to others [ibid., section 3].Hopefully this is enough groundwork for us to move on to chapter 76 of book IV, where St. Thomas begins:
Now, the bestowal of all of these orders accompanies some sacrament, as was said, and the sacraments of the Church require some ministers for their dispensing; there must, therefore, be a superior power in the Church with a higher ministry which dispenses the sacrament of orders. And this is the episcopal power, which, although it does not exceed the power of the priest in the consecration of the body of Christ, does exceed the priestly power in what touches the faithful. For the priestly power itself flows from the episcopal power, and anything particularly difficult to be performed for the faithful is reserved to the bishops; by their authority, even priests are empowered to do that which is committed to them to be done. Hence, even in the tasks which priests perform they employ things consecrated by bishops; thus, in the Eucharistic consecration they use a chalice, an altar, and a pall consecrated by the bishop. Clearly, then, the chief direction of the faithful belongs to the dignity of the bishops [section 1].But it is not enough for us to have bishops. Bishops are the pastors of specific churches in specific places. But if this is our situation, then we necessarily have an obstacle to unity: because bishop may disagree with bishop.
Aquinas says that this is not our situation.
 But this, too, is clear: Although people are set apart according to differing dioceses and states, yet, as the Church is one, so must the Christian people be one. Therefore, as for the specific congregation of one Church one bishop is called for who is the head of that Church; so for the entire Christian people there must be one who is head of the entire Church.Well, what if someone suggests that Christ is the head of the entire Church? Yes, he is, but more must be said.
 Then, too, the unity of the Church requires that all the faithful agree as to the faith. But about matters of faith it happens that questions arise. A diversity of pronouncements, of course, would divide the Church, if it were not preserved in unity by the pronouncement of one. Therefore, the unity of the Church demands that there be one who is at the head of the entire Church. But, manifestly, in its necessities Christ has not failed the Church which He loved and for which He shed His blood, since even of the synagogue the Lord says: ‘What is there that I ought to do more to My vineyard that I have not done to it?” (Isa. 5:4). Therefore, one must not doubt that by Christ’s ordering there is one who is at the head of the entire Church.
 No one should doubt, furthermore, that the government of the Church has been established in the best way, since He has disposed it by whom “kings reign, and lawmakers decree just things” (Prov. 8:15). But the best government of a multitude is rule by one, and this is clear from the purpose of government, which is peace; for peace and the unity of his subjects are the purpose of the one who rules, and one is a better constituted cause of unity than many. Clearly, then, the government of the Church has been so disposed that one is at the head of the entire Church.
 But let one say that the one head and one shepherd is Christ, who is one spouse of one Church; his answer does not suffice. For, clearly, Christ Himself perfects all the sacraments of th Church: it is He who baptizes; it is He who forgives sins; it is He, the true priest, who offered Himself on the altar of the cross, and by whose power His body is daily consecrated on the altar—nevertheless, because He was not going to be with all the faithful in bodily presence, He chose ministers to dispense the things just mentioned to the faithful, as was said above. By the same reasoning, then, when He was going to withdraw His bodily presence from the Church, He had to commit it to one who would in His place have the care of the universal Church. Hence it is that He said to Peter before His ascension: “Feed My sheep” (John 21:17); and before His passion: “You being once converted confirm your brethren” (Luke 22:32); and to him alone did He promise: “I will give to you the keys of the kingdom of heaven” (Mat. 16:19), in order to show that the power of the keys was to flow through him to others to preserve the unity of the Church.The answer does not suffice precisely because the sacraments must be administered visibly, and we must have a visible Church with visible Orders and a visible head. It is not that Christ is not the Head of the Church; of course he is. But the Pope remains the visible head, and serves as Christ's vicar.
 But it cannot be said that, although He gave Peter this dignity, it does not flow on to others. For, clearly, Christ established the Church so that it was to endure to the end of the world; in the words of Isaiah (9:7): “He shall sit upon the throne of David and upon His kingdom to establish and strengthen it with judgment and with justice from henceforth and forever.” It is clear that He so established therein those who were then in the ministry that their power was to be passed on to others even to the end of time; especially so, since He Himself says: “Behold I am with you all days even to the consummation of the world” (Mat. 28:20).
 By this, of course, we exclude the presumptuous error of some who attempt to withdraw themselves from the obedience and the rule of Peter by not recognizing in his successor, the Roman Pontiff, the pastor of the universal Church.