On the Brun--Titchmarsh theorem. II

Denote by $\pi(x;q,a)$ the number of primes $p\leqslant x$ with $p\equiv a\bmod q.$ We prove new upper bounds for $\pi(x;q,a)$ when $q$ is a large prime very close to $\sqrt{x}$, improving upon the classical work of Iwaniec (1982). The proof reduces to bounding a quintilinear sum of Kloosterman sums, to which we introduce a new shifting argument inspired by Vinogradov--Burgess--Karatsuba, going beyond the classical Fourier-analytic approach thanks to a deep algebro-geometric result of Kowalski--Michel--Sawin on sums of products of Kloosterman sums.