We study the convergence rate of Bergman metrics on the class of polarized pointed Kähler n-manifolds (M, L, g, x) with $\text{Vol}\left(B_1(x)\right)>v$ and $\sec \le K$ on M. Relying on Tian’s peak section method (Tian in J Differ Geom 32(1):99–130, 1990), we show that the $C^{1,\alpha }$ convergence of Bergman metrics is uniform. In the end, we discuss the sharpness of our estimates.