Many-body open quantum systems balance internal dynamics against decoherence and measurements induced by interactions with an environment1,2. Quantum circuits composed of random unitary gates with interspersed projective measurements represent a minimal model to study the balance between unitary dynamics and measurement processes3–5. As the measurement rate is varied, a purification phase transition is predicted to emerge at a critical point akin to a fault-tolerant threshold6. Here we explore this purification transition with random quantum circuits implemented on a trapped-ion quantum computer. We probe the pure phase, where the system is rapidly projected to a pure state conditioned on the measurement outcomes, and the mixed or coding phase, where the initial state becomes partially encoded into a quantum error correcting codespace that keeps the memory of initial conditions for long times6,7. We find experimental evidence of the two phases and show numerically that, with modest system scaling, critical properties of the transition emerge.