We examine the cosmic evolution of the growth of perturbations with respect to matter content at early and recent past era of the Universe under the framework of non-zero torsion cosmology. Some cosmographic parameters are also discussed. To analyze this cosmic scenario, we formulate the cosmic models with two dark matter considerations and apply spherical collapse formalism. We explore that for model-1 (pressure less matter), the density contrast starts growing in early era of the Universe and with the Universe's overall expansion, density contrast grows faster for different choices of torsion parameter. In model-2 (matter with non-zero pressure), the density contrast shows enormous growth as compared to overall expansion of the Universe for the perturbed region, which indicates the collapse of the perturbed region and formulation of new large scale matter or galaxy. Further, we analyze the behavior of the growth function for both models with different values of torsion parameter. For model-1, the growth function increases smoothly for different choices of torsion parameter but for the model-2 the behavior of the growth function with nonzero DM and torsion significantly deviates from pressureless DM with zero torsion. We also investigate different cosmographic parameters for both models and analyze their behavior with different choices of torsion parameter and Lambda CDM. The results show the power-law expansion rate of the accelerating Universe with respect to redshift function.