Amply regular graphs are graphs with local distance-regularity constraints. In this paper, we prove a weaker version of a conjecture proposed by Qiao, Park, and Koolen on diameter bounds of amply regular graphs and make new progress on Terwilliger's conjecture on finiteness of amply regular graphs. Terwilliger's conjecture can be considered as a natural extension of the Bannai-Ito conjecture about distance-regular graphs confirmed by Bang, Dubickas, Koolen, and Moulton. As a consequence, we show that there are only finitely many amply regular graphs with parameters $(n,d,α,β)$ satisfying $α\leq 6β-9$. We achieve these results by a significantly improved Lin--Lu--Yau curvature estimate and new Bakry--Émery curvature estimates. We further discuss applications of our curvature estimates to bounding eigenvalues, isoperimetric constants, and expansion properties. In addition, we obtain a volume estimate, which is sharp for hypercubes.