A key open question for transient polymer networks is the molecular reason underlying their elastic stiffening and viscous thickening nonlinearities, with the two prevailing conventional hypotheses being flow-induced (i) increase of the number density of elastically active network strands (i.e., bridging chains) and (ii) nonlinear elastic force-extension chain behavior. The objective of this work is to reliably infer, and theoretically justify, the mechanism driving the nonlinear response of unentangled reversible networks formed by semidilute aqueous solutions of poly(vinyl alcohol) (PVA) and sodium tetraborate (Borax), across a range of 11 compositions and Deborah number varying from similar or equal to 10(-2) to similar or equal to 70 Weakly nonlinear perturbations from equilibrium are quantitatively analyzed via a strain stiffening transient network theory endowed with a single nonlinear parameter and a microscopic statistical mechanical theory for the deformation-induced change of local interchain packing due to conformational anisotropy and its influence on an effective transient cross-link density. As a result, network structuring through stretch-induced enhancement of interchain associations is shown to dominate the leading-order viscous and elastic nonlinearities over the entire range of compositions and frequencies considered. Furthermore, our theoretical approach allows for a quantitative rationalization for the magnitudes of the intrinsic nonlinearities that is in excellent agreement with observations.