In this article, the effect of the pivot point location on the thrust performance of a two-dimensional sinusoidal flapping elliptic airfoil in a forward flight condition is investigated using numerical simulations and in-house water tunnel experiments. On the chord line, three different pivot locations at a distance of 0.25c, 0.5c, and 0.75c from the leading edge of the airfoil are considered, where c is the chord length of the airfoil. The flapping frequency and effective angle of attack are varied to investigate the propulsive performance of the airfoil at a Reynolds number of 5000. It is noticed that a modification in the pivot location significantly influences the linear velocity distribution, the evolution of the leading-edge vortex, and the near wake region on the airfoil. Consequently, both the transient and time-averaged thrust coefficient of the flapping airfoil is considerably affected. In addition, we have observed when the flapping frequency is increased, the time-averaged thrust coefficient of the airfoil tends to increase up to a critical Strouhal number and deteriorates thereafter. The same trend of time-averaged thrust coefficient is seen at all considered pivot locations and effective angle of attacks. Our finding suggests, at the high flapping frequency, the formation of rotation induced adverse suction region around the airfoil and delay in the shedding of the leading edge vortex developed in the previous flapping stroke are the primary sources, attributing to the thrust deterioration of the flapping airfoil with symmetric pivot location 0.5c. On the other hand, the thrust degrading effects at the two asymmetric pivot locations, 0.25c and 0.75c, are triggered by the adverse suction regions induced by asymmetric airfoil-surface velocity distribution as well as airfoil-wake vortices interaction. Moreover, the thrust degradation can be postponed to a higher critical Strouhal number if the airfoil pivot location is set near the leading edge and higher amplitude of effective angle of attack is followed. Besides, we found that the airfoil propulsive efficiency is affected due to a change in the aerodynamic power co-efficient with the modification of the pivot location. Furthermore, our observation concludes that the pivot location at 0.25c from the leading edge has maximum time-averaged thrust and propulsive efficiency performances at least for the range of pivot locations, flapping frequencies, and effective angle of attacks examined here.