As for Papal Supremacy:
There are many places in the Gospels and New Testament that state Peter’s responsibility and authority far beyond just the first place of honor. But also note that Peter is guided by the Holy Spirit and discussions are held before Peter makes a decision which is then communicated and followed. Here are some sources that should be sufficient:
[FONT=Calibri]**Matthew 16: 15-19 **[/FONT][FONT=Times New Roman]He said to them, “But who do you say that I am?” Simon Peter said in reply, “You are the Messiah, the Son of the living God.” Jesus said to him in reply, “Blessed are you, Simon son of Jonah. For flesh and blood has not revealed this to you, but my heavenly Father. And so I say to you, you are Peter, and upon this rock I will build my church,***** and the gates of the netherworld shall not prevail against it. I will give you the keys to the kingdom of heaven.***** Whatever you bind on earth shall be bound in heaven; and whatever you loose on earth shall be loosed in heaven.”[/FONT]
[FONT=Times New Roman][FONT=Arial]ACTS 15: 6 The apostles and the presbyters met together to see about this matter. 7 After much debate had taken place, Peter got up and said to them, “My brothers, you are well aware that from early days God made his choice among you that through my mouth the Gentiles would hear the word of the gospel and believe. . . . [/FONT][/FONT][FONT=Times New Roman][FONT=Arial]24 Since we have heard that some of our number [who went out] without any mandate from us have upset you with their teachings and disturbed your peace of mind, 25 we have with one accord decided to choose representatives and to send them to you along with our beloved Barnabas and Paul, 26 who have dedicated their lives to the name of our Lord Jesus Christ. 27 So we are sending Judas and Silas who will also convey this same message by word of mouth: 28 ‘It is the decision of the holy Spirit and of us not to place on you any burden beyond these necessities, . . .[/FONT][/FONT]
[FONT=Calibri][FONT=Arial]**JOHN 14: *16 And I will ask the Father, and he will give you another Advocate to be with you always, 17 the Spirit of truth, which the world cannot accept, because it neither sees nor knows it. But you know it, because it remains with you, and will be in you. 18 I will not leave you orphans; I will come to you. . . .26 The Advocate, the holy Spirit that the Father will send in my name—he will teach you everything and remind you of all that * told you.
Given that you accept that Jesus Christ is indeed the Living Son of God, and, being God, Christ cannot lie, then what Christ said and did and how the Apostles went about in compliance with Christ’s commands gives us a very clear and strong foundation to believe in the Catholic Church. While we have questions, those questions should never cause us to leave the practice of our Faith.
Given that 1) Christ gave Peter the power to bind and loose and 2) that the Holy Spirit - the Spirit of Truth - was sent from the God the Father, and 3) that the Apostles would meet to discuss and accept Peter’s decisions, it seems pretty clear that on such matters, Peter’s successor - the Pope - is still guided by the Holy Spirit. It is important to distinguished that the Pope’s Infallibility applies to matters of Faith and Morals and not to everything else he may state.
There is still today the Congregation of the Doctrine of Faith at the Vatican that thoroughly researches Faith questions and then the results are presented to the Pope for His decision. Because the Pope is guided by the Hoy Spirit, he alone is responsible for decisions and has the power the accept, modify or change the research. When the Pope does issue a decision, he provides his reasoning with many references to the Bible and past Papal decisions.
Keep in mind that the Holy Spirit is guiding him and he is Christ’s Vicar on Earth. There is no better authority to rely on. History is replete with those who disagree, but disagreement is not a proper basis to overcome the Words of Christ who cannot lie.