The nonequilibrium structural and rheological properties of suspension of rodlike polymers with attractive ends are investigated by mesoscale hydrodynamic simulations. The hybrid simulation approach combines the multiparticle collision dynamics method for the fluid with molecular dynamics simulations for the polymers. In equilibrium, the rodlike polymers self-organize into scaffold-like network structures above a critical density and attraction strength. Shear flow induces significant structural changes. We identify three major regimes. At low Weissenberg numbers, the scaffold structure is compactified accompanied by a phase separation into a fluid domain and a dense scaffold-like structure. The shear viscosity shows a Newtonian plateau, which is determined by the fluid layer. At intermediated shear rates, shear bands appear with bundles of polymers. Again, we observe a nearly Newtonian viscosity regime governed by the fluid bands. At high shear rates, any structure is dissolved and a paranematic phase appears, with rods well aligned with the flow, and shear thinning appears.