Neural-network-based algorithms have garnered considerable attention for their ability to learn complex patterns from very-high-dimensional data sets towards classifying complex long-range patterns of entanglement and correlations in many-body quantum systems, and towards processing high-dimensional classical data sets. Small-scale quantum computers are already showing potential gains in learning tasks on large quantum and very large classical data sets. A particularly interesting class of algorithms, the quantum convolutional neural networks (QCNNs) could learn features of a quantum data set by performing a binary classification task on a nontrivial phase of quantum matter. Inspired by this promise, we present a generalization of QCNN, the “branching quantum convolutional neural network,” or bQCNN, with substantially higher expressibility. A key feature of bQCNN is that it leverages midcircuit (intermediate) measurement results, realizable on several current quantum devices, obtained in pooling layers to determine which sets of parameters will be used in the subsequent convolutional layers of the circuit. This results in a “branching” structure, which allows for a greater number of trainable variational parameters in a given circuit depth. This is of particular use in current-day noisy intermediate-scale quantum devices, where circuit depth is limited by gate noise. We present an overview of the Ansatz structure and scaling and provide evidence of its enhanced expressibility compared with QCNN. Using artificially constructed large data sets of training states as a proof of concept, we demonstrate the existence of training tasks in which bQCNN far outperforms an ordinary QCNN. We provide an explicit example of such a task in the recognition of the transition from a symmetry protected topological to a trivial phase induced by multiple, distinct perturbations. Finally, we present future directions where the classical branching structure and increased density of trainable parameters in bQCNN would be particularly valuable.