We study the entanglement dynamics and the possible phase transitions in a generic quantum automaton circuit subjected to projective measurements. We design an efficient algorithm which not only allows us to perform large-scale simulation for the Rényi entropy, but also provides a physical picture for the entanglement dynamics, which can be interpreted in terms of a classical bit-string model which belongs to the directed percolation universality class. We study the purification dynamics of a state formed by Einstein-Podolsky-Rosen pairs, and the growth of entanglement starting from a product state. In both cases, we verify numerically that the dynamics is in the universality class of classical directed percolation.