Due to its finite size and the large centrifugal load, the fir-tree root is highly stressed, which leads to the possible early failure of the gas turbine and steam turbine. To find an optimized fir-tree root is an important issue for the design of the turbine structures. In this paper, a superellipse-based design optimization approach is proposed for the fir-tree root. Rather than the straight line and arc used in literature, the combination of the superellipse curve and line are employed to characterize the fir-tree root since the superellipse curve represents a large family of curves with limited parameters, which makes the design optimization easy and economic. For the design optimization, the objective function is to minimize the peak stress, which is a typical min-max problem with possible severe iterative oscillation and subsequent convergence difficulty. To avoid this problem, a P-norm aggregation function is proposed. The superellipse parameters are defined as design variables, while the stress concentration factor and the stress at root neck are specified as optimization constraints. With the P-series fir-tree root design as example, it is proved that our approach is effective to find the optimized configuration with better stress distribution and lower stress concentration.