Distinct channels of interaction in a complex networked system define network layers, which co-exist and co-operate for the system's function. Towards realistic modeling and understanding such multiplex systems, we introduce and study a class of growing multiplex network models in which different network layers coevolve, and examine how the entangled growth of coevolving layers can shape the overall network structure. We show analytically and numerically that the coevolution can induce strong degree correlations across layers, as well as modulate degree distributions. We further show that such a coevolution-induced correlated multiplexity can alter the system's response to dynamical process, exemplified by the suppressed susceptibility to a threshold cascade process.