We introduce a new local-to-global structure learning algorithm, called graph growing structure learning (GGSL), to learn Bayesian network (BN) structures. GGSL starts at a (random) node and then gradually expands the learned structure through a series of local learning steps. At each local learning step, the proposed algorithm only needs to revisit a subset of the learned nodes, consisting of the local neighborhood of a target, and therefore improves on both memory and time efficiency compared to traditional global structure learning approaches. GGSL also improves on the existing local-to-global learning approaches by removing the need for conflict-resolving AND-rules, and achieves better learning accuracy. We provide theoretical analysis for the local learning step, and show that GGSL outperforms existing algorithms on benchmark datasets. Overall, GGSL demonstrates a novel direction to scale up BN structure learning while limiting accuracy loss.