We explore "hybrid" quantum circuit models in one-dimension with both unitary and measurement gates and investigate the entanglement properties of wavefunction trajectories at long times, in the steady state.  We find a volume law entangled phase for low measurement rates, which exhibits a sub-dominant logarithmic behavior in the entanglement entropy. With increasing measurement rate, the volume law phase is unstable to a disentangled area law phase, passing through a single entanglement transition at a critical rate of measurement. At criticality, we further establish the emergence of conformal field theories in (1+1)-dimensional quantum circuits, by revealing space-time conformal covariance of entanglement entropies and mutual information for various subregions at different circuit depths.