A magnetized nonthermal electron–positron-ion (e-p-i) plasma is considered to study the propagation properties of ion-acoustic solitary and shock waves in the presence of trapped positrons and electrons for the first time. The Schamel-κ (kappa) distribution function that describes plasma nonthermality and particle trapping is assumed to consider electrons and positrons. The diffusive effect of ion plasma fluid, which is responsible for shock dynamics, is taken into account. A nonlinear Schamel-Korteweg–de Vries-Burgers' (SKdVB) equation is derived by employing the reductive perturbation approach, and the solitary and shock wave solutions of the SKdVB equation have also been derived for different limiting cases. It is found that only positive potential nonlinear structures (for both solitary and shock waves) are formed in the proposed plasma system. The condition for stable solitons in the absence of dissipation is analyzed, and the nature of arbitrary amplitude solitary waves (obtained via the Sagdeev potential approach) is discussed. It is found through theoretical and numerical investigation that different plasma compositional parameters (such as the trapping effect of electrons (βe) and positrons (βp), the obliquity effect (θ), electron-to-ion number density ratio (µe), the magnetic field effect (via Ω) and the viscous effect (via η)) have a significant influence on the dynamics of ion-acoustic solitary and shock waves. The theoretical and numerical investigations in this study may be helpful in describing the nature of localized structures in different plasma contexts, e.g. space and astrophysical plasmas and experimental plasmas where electron–positron-ion plasmas exist.