"Between 313 and 590 the Old Catholic church, in which each bishop had been an equal, became the Roman Catholic church, in which the bishop of Rome won primacy over the other bishops." -- Earle Cairns, 157.
Starting out, the Church had many bishops who governed different geographical regions of Christendom. Five bishops had the most leverage: Constantinople, Antioch, Jerusalem, Alexandria, and Rome. When Constantine made Constantinople the capital, the political attention moved east, but the Roman bishop had the most influence in the west. Also, the eastern empire struggled with heresies such as Arianism, Donatism, and other isms. Rome did not have such divisions in the west. Therefore, the Roman bishop was the most theologically sound bishop in the known world.
So, the people in Rome began to see its bishop as a first among equals. They assumed that their bishop was in a line of succession from Peter based on Matthew 16:16-18, Luke 22:31-32, and John 21:15-17. This was fully accepted by 590. At that point, Rome still had good theology. She had giants like Augustine and Ambrose who fought for truth in the empire even to the extent of rebuking popes. Even the missionaries would create converts and tell them to yield allegiance to the bishop of Rome. Leo I was the first Roman bishop to view himself as the top bishop in the world. Gregory the Great was the first one to really use the title "pope." When barbarians attacked the empire, the popes were the ones who led them n battle and defeated the Vandals and Goths, etc.
Such faith in the Roman bishop began a downward spiral of humans sitting in Christ's seat and proclaiming to be the ultimate authority in the Church, even when they contradict the Bible. At that time, most people could not read a Bible, so the Church did need men to interpret for the people and to lead them in worship. However, Christ is the only head of the Church. His human representative is the whole Church that worships Him, not one guy in Rome. Christ is also the only source of special revelation, the Bible, not the Church or the Pope. This experiment started out okay but then began to focus on just one person at the expense of what the Bible actually said. At this point, the pope was still humble enough to accept instruction from Augustine and the like, but it is never a good thing to place your faith in one human that is not Christ, no matter how honorable.