In this study, we explore the dynamics of Bianchi type-III space-time within the framework of $f(T)$ gravity, focusing on both linear and non-linear forms of $f(T)$ function. We analyze the behavior of cosmological parameters by assuming the deceleration parameter (DP) as a simple linear function of the Hubble parameter. Key cosmological parameters such as the scale factor, Hubble parameter, DP, spatial volume, shear scalar, expansion scalar, energy density, pressure, and the equation of state (EoS) parameter are expressed in terms of the redshift parameter. Their dynamic behaviors are graphically presented for both linear and non-linear forms of $f(T)$ gravity. Our results align with recent cosmological observations, with the non-linear form of $f(T)$ exhibiting a stronger tendency toward accelerated cosmic expansion compared to the linear model. The EoS parameter indicates a quintessence phase, driving the universe’s accelerated expansion, as recently investigated by Varshney et al. (Can. J. Phys. 102(3):199–209, 2023). Additionally, we examine the violation of the strong energy conditions, a crucial aspect in modified gravity theories. The model parameter $\xi $ and the current value of the Hubble parameter $H_{0}$ are estimated using the Hubble data set and Pantheon+ SHOES data set, further validating our theoretical model.
