Majority dynamics on a graph G is a deterministic process such that every vertex updates its ±1-assignment according to the majority assignment on its neighbor simultaneously at each step. Benjamini, Chan, O'Donnell, Tamuz and Tan conjectured that, in the Erdős-Rényi random graph G(n, p), the random initial ±1-assignment converges to a 99%-agreement with high probability whenever p = 𝜔(1∕n). This conjecture was first confirmed for p ≥ 𝜆n −1∕2 for a large constant 𝜆 by Fountoulakis, Kang and Makai. Although this result has been reproved recently by Tran and Vu and by Berkowitz and Devlin, it was unknown whether the conjecture holds for p < 𝜆n −1∕2 . We break this Ω(n −1∕2 )-barrier by proving the conjecture for sparser random graphs G(n, p), where 𝜆 ′ n −3∕5 log n ≤ p ≤ 𝜆n −1∕2 with a large constant 𝜆 ′ > 0.