Quasi-one-dimensional fibril crystal growth of diblock copolymers is a fundamental issue in the investigation of nanotechnology and neurodegenerative diseases. We performed dynamic Monte Carlo simulations of lattice polymers to study the crystallization-driven fibril crystal growth of diblock copolymers under two circumstances of solutions: sporadic crystals with the feeding mode of constant polymer concentrations, and massive crystals with the depleting mode of decaying polymer concentrations. We confirmed anisotropic driving forces as a prerequisite of steady growth of fibril crystals. The lamellar crystal width is confined by the noncrystalline block below a critical concentration that shifts down with the decrease of the noncrystallizable block fractions. In the depleting mode, the long-axis sizes at the early stage of fibril crystal growth can be fitted well by an exponential-decay function of time, and the growth rates decrease linearly with polymer concentrations by following the growth rates in the feeding mode, appearing as consistent with our previous simulation results of homopolymer solutions.