1. Christ does not say that He is making Peter the head of His church.
2. The key to understanding this scripture lies in the correct translation of the Greek words here rendered "Peter" and "rock." The Greek word translated "Peter" is petros, meaning "pebble" or "small stone," while the word rendered "rock" is petra, meaning "big rock" or "huge boulder."
3. Christ says He would build His church on the boulder, not on Peter the pebble. Christ is the rock (I Corinthians 10:4). He is also the "chief cornerstone," upon which the church is built (Ephesians 2:20).
4. Although Peter is shown in a leadership position among the apostles throughout the gospels and Acts, the early church leaders did not function in a strict hierarchical manner. Read and study Acts 15:1-29. Here is a ministerial conference called to discuss a matter of doctrine causing division in the church (verse 6). Peter makes his point (verses 7-11), which is later adopted by the other apostles and elders (verse 22). But it is James, the physical brother of Jesus Christ and pastor of the Jerusalem church, who sums up the conference's decision (verses 13-19). This helps us to understand how the apostles (plural) form the foundation of the church with the prophets (Ephesians 2:20).
5. Peter was not infallible. Read Matthew 16:21-23, where Jesus has to rebuke Peter severely for a wrong attitude immediately after His statement about the "rock." In addition, the apostle Paul later corrects him publicly for returning to the Jewish practice of refusing to eat with Gentiles (Galatians 2:11-14).
From these proofs, it is easy to see that Jesus did not say He would build His church on Peter, a mere man, but on Himself, and because of that, the church would endure and prevail.