Simulating the dynamics of complex quantum systems is a central application of quantum devices. Here we propose leveraging the power of measurements to simulate short-time quantum dynamics of physically prepared quantum states in classical postprocessing using a truncated Taylor series approach. While limited to short simulation times, our hybrid quantum-classical method is equipped with rigorous error bounds. It is extendable to estimate low-order Taylor approximations of smooth time-dependent functions of tractable linear combinations of measurable operators. These insights can be made use of in the context of Hamiltonian learning and device verification, short-time imaginary-time evolution, or the application of intractable operations to subuniversal quantum simulators in classical postprocessing.