In this paper, we study the practical exponential stability (PES) of nonlinear time-delay (TD) systems with bounded exogenous input (BEI). Unlike existing results, this work is the first to provide a sum-of-squares (SOS)-based design to ensure the PES of the nonlinear TD system. First, polynomial fuzzy (PF) modeling is used to characterize the nonlinear behavior of the system. Specifically, we consider a class of models in which the premise variables are measurable, and the polynomial matrices depend on any measurable variable. Second, we propose a PF observer-based (PFOB) control strategy that guarantees the system's PES under the condition that the system is observable, even when some state variables are not directly measurable. Next, the PFOB controller is designed via the SOS approach to ensure the PES of the augmented system formed by the state and the state estimation error. This study is the first to incorporate recently proposed relaxed conditions for parameterized linear matrix inequalities formulated in a double-sum representation into the SOS approach. Finally, a numerical case study validates the PFOB control strategy.