Compliant mechanisms’ design aims to create a larger workspace and simple structural shapes because these mechanical systems usually have small dimensions, reduced friction, and less bending. From that request, we designed optimal bridge-type compliant mechanism flexure hinges with a high magnification ratio, low stress by using a flexure joint, and especially no friction and no bending. This joint was designed with optimal dimensions for the studied mechanism by using the method of grey relational analysis (GRA), which is based on the Taguchi method (TM), and finite element analysis (FEA). Grey relational grade (GRG) has been estimated by an artificial neural network (ANN). The optimal values were in good agreement with the predicted value of the Taguchi method and regression analysis. The finite element analysis, signal-to-noise analysis, surface plot, and analysis of variance demonstrated that the design dimensions significantly affected the equivalent stress and displacement. The optimal values of displacement were also verified by the experiment. The outcomes were in good agreement with a deviation lower than 6%. Specifically, the displacement amplification ratio was obtained as 65.36 times compared with initial design.