This article presents a novel approach to the safety verification of hybrid systems by synthesizing neural barrier certificates (BCs) via counterexample-guided neural network (NN) learning combined with sum-of-square (SOS)-based verification. We learn more easily verifiable BCs with NN polynomial expansions in a high-accuracy counterexamples guided framework. By leveraging the polynomial candidates yielded from the learning phase, we reformulate the identification of real BCs as convex linear matrix inequality (LMI) feasibility testing problems, instead of directly solving the inherently NP-hard nonconvex bilinear matrix inequality (BMI) problems associated with SOS-based BC generation. Furthermore, we decompose the large SOS verification programming into several manageable subprogrammings. Benefiting from the efficiency and scalability advantages, our approach can synthesize BCs not amenable to existing methods and handle more general hybrid systems.