A many-qubit system subject to random unitary evolution with interspersed projective measurements can exhibit distinct entangling and disentangling phases, separated by a critical measurement rate. However, these systems have been studied primarily using classical calculations with Clifford circuits. Here, we report a study of measurement-induced entanglement transitions using transmon-based IBM Quantum devices, which support mid-circuit measurements and sub-microsecond readout times. We observe a crossover in the entanglement entropy between an entangling phase, characterized by volume-law scaling, and a disentangling phase, characterized by area-law scaling. We further investigate crossovers induced by weak measurements of varying strength, using ancillary qubits. Lastly, we estimate the critical measurement rate and critical exponents characterizing the transition. Our work indicates that near-term quantum computers can be useful in exploring quantum phases and dynamical behaviour under monitoring protocols. *S. N. S. and A. J. M. were supported by NSF under Award No. 1839204.