Owing to improvements in image recognition via deep learning, machine-learning algorithms could eventually be applied to automated medical diagnoses that can guide clinical decision-making. However, these algorithms remain a 'black box' in terms of how they generate the predictions from the input data. Also, high-performance deep learning requires large, high-quality training datasets. Here, we report the development of an understandable deep-learning system that detects acute intracranial haemorrhage (ICH) and classifies five ICH subtypes from unenhanced head computed-tomography scans. By using a dataset of only 904 cases for algorithm training, the system achieved a performance similar to that of expert radiologists in two independent test datasets containing 200 cases (sensitivity of 98% and specificity of 95%) and 196 cases (sensitivity of 92% and specificity of 95%). The system includes an attention map and a prediction basis retrieved from training data to enhance explainability, and an iterative process that mimics the workflow of radiologists. Our approach to algorithm development can facilitate the development of deep-learning systems for a variety of clinical applications and accelerate their adoption into clinical practice.