Entanglement properties of purified quantum states are of key interest for two reasons. First, in quantum information theory, minimally entangled purified states define the entanglement of purification as a fundamental measure for the complexity of the corresponding physical mixed state. Second, dynamical entanglement growth in purified states represents the main bottleneck for calculating dynamical physical properties on classical computers in the framework of tensor network states. Here, we demonstrate how geometric methods, including parallel transport, may be harnessed to reduce such dynamical entanglement growth, and how to obtain a general prescription for maintaining (locally) optimal entanglement entropy when time-evolving a purified state. Adapting and extending by higher-order skew corrections the notion of Uhlmann geometric phases, we reveal the relation between dynamical entanglement growth and the geometry of the Hilbert-Schmidt bundle as the mathematical foundation of purified states. With benchmarks on a nonintegrable spin chain model, we compare the computational performance of matrix product state algorithms based on our present geometric disentangling method to previous approaches for taming entanglement growth in purified states. Our findings provide numerical evidence that geometric disentanglers are a powerful approach, superior in various aspects to known methods for disentangling purified states in a range of physically relevant computational scenarios. To exclude the effect of algorithmic imperfections, we also provide a numerically exact analysis for systems of moderate size.