Visualizations have become crucial in the contemporary data-driven world as they aid in exploring, verifying, and sharing insights obtained from data. In this paper, we propose a new paradigm of visualization synthesis based on refinement types. Besides input-output examples, users can optionally use refinement-type annotations to constrain the range of valid values in the example visualization or to express complex interactions between different visual components. Our system's outputs include both data transformation and visualization programs that are consistent with refinement-type specifications. To mitigate the scalability challenge during the synthesis process, we introduce a new visualization synthesis algorithm that uses lightweight bidirectional type checking to prune the search space. As we demonstrate experimentally, this new synthesis algorithm results in significant speed-up compared to prior work.