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We theoretically investigate the out-of-equilibrium dynamics in binary Bose-Einstein condensate confined within two-dimensional box potentials. One species of the condensate interacts with a pair of oppositely wound, but otherwise identical Laguerre-Gaussian laser pulses, while the other species is influenced only via the interspecies interaction. Depending on the number of helical windings (or the magnitude of topological charge), the species directly participating in the interaction with lasers is dynamically segmented into distinct parts that collide together as the pulses gradually diminish. This collision event generates nonlinear structures in the related species, coupled with the complementary structures produced in the other species, due to the interspecies interaction. The long-time dynamics of the optically perturbed species is found to develop the Kolmogorov-Saffman scaling law in the incompressible kinetic energy spectrum, a characteristic feature of quantum turbulence. This study warrants the usage of Laguerre-Gaussian beams for future experiments on quantum turbulence in Bose-Einstein condensates. • We show the out-of-equilibrium dynamics of a binary Bose-Einstein condensate interacting with two Laguerre-Gaussian (LG) beams. • Condensate is segregated into multiple segments depending on the charge of LG beams. • Collision of the segregated segments results in the generation of vortex and antivortex. • We observe −5/3 and −4 power-law in the incompressible kinetic energy spectra. [ABSTRACT FROM AUTHOR]