Interpretability for neural networks is a trade-off between three key requirements: 1) faithfulness of the explanation (i.e., how perfectly it explains the prediction), 2) understandability of the explanation by humans, and 3) model performance. Most existing methods compromise one or more of these requirements; e.g., post-hoc approaches provide limited faithfulness, automatically identified feature masks compromise understandability, and intrinsically interpretable methods such as decision trees limit model performance. These shortcomings are unacceptable for sensitive applications such as education and healthcare, which require trustworthy explanations, actionable interpretations, and accurate predictions. In this work, we present InterpretCC (interpretable conditional computation), a family of interpretable-by-design neural networks that guarantee human-centric interpretability, while maintaining comparable performance to state-of-the-art models by adaptively and sparsely activating features before prediction. We extend this idea into an interpretable, global mixture-of-experts (MoE) model that allows humans to specify topics of interest, discretely separates the feature space for each data point into topical subnetworks, and adaptively and sparsely activates these topical subnetworks for prediction. We apply variations of the InterpretCC architecture for text, time series and tabular data across several real-world benchmarks, demonstrating comparable performance with non-interpretable baselines, outperforming interpretable-by-design baselines, and showing higher actionability and usefulness according to a user study.