Neural network-based algorithms have garnered considerable attention in condensed matter physics 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. 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 (QCNN) 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 mid-circuit (intermediate) measurement results, realizable on current trapped-ion systems, 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 on current-day NISQ 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 to 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. Finally, we present future directions where the classical branching structure and increased density of trainable parameters in bQCNN would be particularly valuable.