No Contradiction in Matthew

No. Actually, the passages are harmonious. In Matthew 16:18-19, Jesus makes clear that Peter is the visible head of his Church. And in Matthew 20:25-26, Jesus explains to Peter and the other apostles what kind of leaders they should be. That is, regarding their authority, they should not “lord it over” their brothers and sisters, as do Gentile rulers. Instead, they should have the attitude of a servant in exercising his authority, looking out for the genuine good of those whom he serves.

Consequently, one of the titles popes have long embraced is “Servant of the Servants of God.”