Viral blips are a recurrent pattern observed in many persistent infections such as the human immunodeficiency virus (HIV). The main goal of this research is to present a comprehensive analytical study of a two-dimensional discrete-time in-host infection model, that exhibits viral blips, with a saturating infection rate. We examine the interactions between the population densities of infected and uninfected CD4+ T cells by discussing the model's dynamics in the long run. The local asymptotic stability of fixed points of the model is investigated. The model undergoes both flip and Neimark–Sacker bifurcations. Moreover, codimension-two bifurcations of the endemic fixed point are discussed using bifurcation theory and normal forms. The model exhibits 1:2, 1:3, and 1:4 strong resonances. Numerical simulations are performed to verify our analysis. In addition, bifurcations of higher iterations are extracted from the numerical continuation.