The transient response of associative copolymer solutions during step shear rate experiments has been characterized. Upon cessation of flow, the stress relaxation modulus is found to decrease faster as the shear rate increases. Upon start up of the flow, beyond a critical value of the shear rate, one or several stress overshoots are observed, depending on the shear rate value. One of them corresponds to the stress overshoot characteristic of entangled polymer systems while the other ones are specific to the presence of hydrophobic units. The critical shear rate and the accumulated strain position that characterize the appearance of the overshoots can be most easily explained if the solution is viewed as a loosely connected transient network. Finally, aging phenomena are also described and are tentatively ascribed to a trend to phase separation.