Well, I guess it would be an arrogant statement if it were merely my opinion.
It's true that the Bible does not say, "Then Jesus started the Catholic Church." But it does say in Matthew 16:16-18 that Jesus changed Simon's name to Peter (which means "Rock") and said he would build his church upon this rock. Pretty plain as day to me that Jesus appointed Peter as the leader of his church, unless you do a mental one-handed backwards somersault to explain this passage to mean something else.
Peter handed down his authority to Linus, who handed down his authority to Cletus, who handed his authority down to Clement...and it happened this way until we get to Francis.
Peter was our first Pope. They didn't call him "Pope" at the time. "Pope" is a title that was coined later to indicate the head of the bishops.
But can we know this is true? I mean, it's not in the Bible!
Catholics have a different understanding of what a bishop is compared to the average independent non-denominational local church in America (I say "America" because only in America do we have small churches who do what they want with no authority to submit to. Could be the product of the "me" mentality of America drifting into our churches, I dunno. We can chase that rabbit later...) How can we know whose interpretation of the role of a bishop is correct?
We can look back at what the early church writers had to say about bishops. These writers were the apostles of the apostles. And then the apostles of the apostles of the apostles. In most cases these were the men who either knew the apostles personally or knew someone who knew the apostles (as in the apostles were a sort of a spiritual great-grandfather to them).
What did they write about bishops in the early church back then?
Ignatius of Antioch
Follow your bishop, every one of you, as obediently as Jesus Christ followed the Father. Obey your clergy too as you would the apostles; give your deacons the same reverence that you would to a command of God. Make sure that no step affecting the Church is ever taken by anyone without the bishop’s sanction. The sole Eucharist you should consider valid is one that is celebrated by the bishop himself, or by some person authorized by him. Where the bishop is to be seen, there let all his people be; just as, wherever Jesus Christ is present, there is the Catholic Church (Letter to the Smyrneans 8:2 [A.D. 110]).
In like manner let everyone respect the deacons as they would respect Jesus Christ, and just as they respect the bishop as a type of the Father, and the presbyters as the council of God and college of the apostles. Without these, it cannot be called a Church. I am confident that you accept this, for I have received the exemplar of your love and have it with me in the person of your bishop. His very demeanor is a great lesson and his meekness is his strength. I believe that even the godless do respect him (Letter to the Trallians 3:1-2 [A. D. 110]).
The Catholic Church possesses one and the same faith throughout the whole world, as we have already said (Against Heresies 1:10 [A.D. 189]).
Since therefore we have such proofs, it is not necessary to seek the truth among others which it is easy to obtain from the Church; since the apostles, like a rich man [depositing his money] in a bank, lodged in her hands most copiously all things pertaining to the truth: so that every man, whosoever will, can draw from her the water of life. For she is the entrance to life; all others are thieves and robbers. On this account we are bound to avoid them, but to make choice of the things pertaining to the Church with the utmost diligence, and to lay hold of the tradition of the truth. For how stands the case? Suppose there should arise a dispute relative to some important question among us. Should we not have recourse to the most ancient churches with which the apostles held constant intercourse, and learn from them what is certain and clear in regard to the present question? For how should it be if the apostles themselves had not left us writings? Would it not be necessary [in that case] to follow the course of the tradition which they handed down to those to whom they did commit the churches? (ibid. 3:4).
Where was Marcion then, that shipmaster of Pontus, the zealous student of Stoicism? Where was Valentinus then, the disciple of Platonism? For it is evident that those men lived not so long ago – in the reign of Antoninus for the most part – and that they at first were believers in the doctrine of the Catholic Church, in the church of Rome under the episcopate of the blessed Eleutherus, until on account of their ever restless curiosity, with which they even infected the brethren, they were more than once expelled (On the Prescription Against Heretics 22,30 [A.D.200])
But if there be any (heresies) which are bold enough to plant themselves in the midst Of the apostolic age, that they may thereby seem to have been handed down by the apostles, because they existed in the time of the apostles, we can say: Let them produce the original records of their churches; let them unfold the roll of their bishops, running down in due succession from the beginning in such a manner that [that first bishop of theirs] bishop shall be able to show for his ordainer and predecessor some one of the apostles or of apostolic men,--a man, moreover, who continued stedfast with the apostles. For this is the manner in which the apostolic churches transmit their registers: as the church of Smyrna, which records that Polycarp was placed therein by John; as also the church of Rome, which makes Clement to have been ordained in like manner by Peter. (On the Prescription Against Heretics 22,30 [A.D.200])Cyprian
The spouse of Christ cannot be defiled; she is uncorrupted and chaste. She knows one home . . . Does anyone believe that this unity which comes from divine strength, which is closely connected with the divine sacraments, can be broken asunder in the Church and be separated by the divisions of colliding wills? He who does not hold this unity, does not hold the law of God, does not hold the faith of the Father and the Son, does not hold life and salvation (On the Unity of the Catholic Church 6 [A.D. 251]).
Peter speaks there, on whom the Church was to be built, teaching and showing in the name of the Church, that although a rebellious and arrogant multitude of those who will not hear or obey may depart, yet the Church does not depart from Christ; and they are the Church who are a people united to the priest, and the flock which adheres to its pastor. Whence you ought to know that the bishop is in the Church, and the Church in the bishop; and if any one be not with the bishop, that he is not in the Church, and that those flatter themselves in vain who creep in, not having peace with God’s priests, and think that they communicate secretly with some; while the Church which is Catholic and one, is not cut nor divided, but is indeed connected and bound together by the cement of priests who cohere with one another (Letters 66 [A.D. 253]).I love reading this stuff. History is fascinating to me, and I can get lost to the world when I read this stuff. You can read more for yourself at Early Christian Writings. The writings are listed by the approximate date they were written, and there is a rating for each as to the reliability of the dating of each. Some writings are "more sturdy" than others. Some of the important writings in church history are from Polycarp, Tertullian, Irenaeus, Origen, Ignatius of Antioch. The entire collection of writings are at this website, not just bits and pieces and quotes.
These writings cover a multitude of topics, not just about bishops. And please do not misunderstand me - I do not equate these writings with the Bible. The Bible is the inerrant Word of God. These writings are merely historical information.
Peace and love to you,