We present an experimental study of the rheology and structure of a confined suspension of deformable particles flowing in a quasi-two-dimensional Poiseuille flow. Thanks to a precise microfluidic viscosimetry technique combined with measurements of concentration profiles, our study provides the first experimental confirmation with three-dimensional particles of a strong relationship between structuring effects and rheology, previously only reported in numerical simulations of purely two-dimensional systems. In conditions where strong structuring effects take place due to confinement, the evolution of the effective viscosity with particle concentration (here, red blood cells) shows a remarkable succession of ranges of rapid growth and plateaus that are associated to qualitative transitions in the structure of the suspension.