An artificial tangential velocity is introduced into evolving finite element methods for mean curvature flow, Willmore flow, and flow driven by a prescribed smooth velocity field, aiming to enhance mesh quality during computations. The construction of this artificial tangential velocity is based on a limiting case of the method proposed by Barrett, Garcke, and Nürnberg (J. Comput. Phys. 222(1), 441-467, 2007). We establish the stability of the artificial tangential velocity and prove optimal-order convergence for the evolving finite element methods applied to mean curvature flow, Willmore flow, and flow under a given velocity field. Extensive numerical experiments are provided, demonstrating the convergence of the methods, the effectiveness of the artificial tangential velocity in improving mesh quality, and significant improvements in the accuracy of solving PDEs on evolving domains.