Learning low-dimensional representations of bipartite graphs enables e-commerce applications, such as recommendation, classification, and link prediction. A layerwise-trained bipartite graph neural network (L-BGNN) embedding method, which is unsupervised, efficient, and scalable, is proposed in this work. To aggregate the information across and within two partitions of a bipartite graph, a customized interdomain message passing (IDMP) operation and an intradomain alignment (IDA) operation are adopted by the proposed L-BGNN method. Furthermore, we develop a layerwise training algorithm for L-BGNN to capture the multihop relationship of large bipartite networks and improve training efficiency. We conduct extensive experiments on several datasets and downstream tasks of various scales to demonstrate the effectiveness and efficiency of the L-BGNN method as compared with state-of-the-art methods. Our codes are publicly available at https://github.com/TianXieUSC/L-BGNN.