I don't see any shocks on this thing.
Women drivers... bouncing up, it seems. The horses' back legs look so close to the wheels.
Many private chariots had a rudamentary suspension that isolated the passenger area from the suspension via leather straps. Others used leather mesh in place of a hard floor to cushion a standing rider. Some combined both.
IIRC, fixed suspension chariots were pretty much only used in warfare or racing (where unpredictable bouncing could be dangerous). The ride in a leather-shocked chariot wouldn't have been perfectly smooth, but it wouldn't have jarred any teeth loose either.