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We present random quantum circuit models for nonunitary quantum dynamics of free fermions in one spatial dimension. Numerical simulations reveal that the dynamics tends toward steady states with logarithmic violations of the entanglement area law and power law correlation functions. Moreover, starting with a short-range entangled many-body state, the dynamical evolution of entanglement and correlations quantitatively agrees with the predictions of two-dimensional conformal field theory with a spacelike time direction. We argue that this behavior is generic in nonunitary free quantum dynamics with time-dependent randomness, and we show that the emergent conformal dynamics of two-point functions arises out of a simple “nonlinear master equation.”