The viscosity enhancement of a solvent produced by the addition of thickening branched polymers is predicted as a function of polymer concentration, branch length and persistence length, and strength of the covalent bonding interactions. Nonequilibrium, stationary-state Poiseuille numerical simulations are performed using the dissipative particle dynamics model to obtain the viscosity of the fluid. It is found that the clustering of the polymers into aggregates increases the viscosity and that it is more strongly affected by the strength of the bonding interactions. General scaling relationships are found for the viscosity as a function of the variables studied, which are expected to be useful for the design and synthesis of new viscosifying polymers. It is argued that our results can be applied to aqueous thickeners, of importance for colloidal fluids such as paints and coatings and also for nonpolar fluids such as supercritical CO2, which is a promising nonhydraulic fracking fluid also useful in enhanced oil recovery.