Abstract The theory of limits of permutations leads to limit objects called permutons, which are certain Borel measures on the unit square. We prove that permutons avoiding a given permutation of order have a particularly simple structure. Namely, almost every fiber of the disintegration of the permuton (say, along the x‐axis) consists only of atoms, at most many, and this bound is sharp. We use this to give a simple proof of the “permutation removal lemma.”