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[Display omitted] • Systematic investigation of the linear and nonlinear shear rheology of a well-characterized moderately entangled H-polystyrene having branches and backbone of nearly the same molar mass. The polymer is checked to be structurally pure via TGIC. • The nonlinear shear experiments are performed with state-of-the-art rheometry using the cone-partitioned plate fixture. • The BoB model describes the linear viscoelasticity as well as the nonlinear start-up shear response, with overshoot and steady state, at lower shear rates. • The presented dataset is useful for further analysis and comparisons in the direction of extending the BoB model by including finite extensibility effects. We present a systematic investigation of the linear viscoelasticity and nonlinear start-up shear flow response of a well-characterized H-polystyrene, an archetype branched polymer. The Branch-on-Branch (BoB) model describes the linear response well, as well as the nonlinear transient response at lower rates, in the regime where finite extensibility effects do not seem to be important. The data reveal shear thinning similar to that of comb polymers and a broad transient overshoot of the shear stress growth coefficient, which suggests a coupled response of backbone and branches. The rich features of the data and comparison with the BoB-model should provide a useful framework for further exploring the transient molecular response of branched polymers in complex flows. [ABSTRACT FROM AUTHOR]