State-of-the-art LSTM language models trained on large corpora learn sequential contingencies in impressive detail and have been shown to acquire a number of non-local grammatical dependencies with some success. Here we investigate whether supervision with hierarchical structure enhances learning of a range of grammatical dependencies, a question that has previously been addressed only for subject-verb agreement. Using controlled experimental methods from psycholinguistics, we compare the performance of word-based LSTM models versus two models that represent hierarchical structure and deploy it in left-to-right processing: Recurrent Neural Network Grammars (RNNGs) (Dyer et al., 2016) and a incrementalized version of the Parsing-as-Language-Modeling configuration from Charniak et al. (2016). Models are tested on a diverse range of configurations for two classes of non-local grammatical dependencies in English-Negative Polarity licensing and Filler-Gap Dependencies. Using the same training data across models, we find that structurally-supervised models outperform the LSTM, with the RNNG demonstrating best results on both types of grammatical dependencies and even learning many of the Island Constraints on the filler-gap dependency. Structural supervision thus provides data efficiency advantages over purely string-based training of neural language models in acquiring human-like generalizations about non-local grammatical dependencies.