Abstract: With recent progress on optimized stellarator designs leading to reduced neoclassical transport and energetic particle prompt loss, there is a pressing need to evaluate the turbulent transport in these stellarators and to incorporate optimization of turbulent transport in the reactor design. Turbulent transport caused by ion temperature gradient (ITG) and trapped electron mode (TEM) instabilities play a major role in transport processes and strongly affects confinement properties. Given the complexity of three-dimensional magnetic fields, global gyrokinetic simulations are crucial for determining the global ITG and TEM turbulent transport level and resulting confinement property in these optimized stellarators. Here we present global gyrokinetic GTC simulations of electrostatic ITG and TEM turbulence in recently proposed quasi-axisymmetric (QA) stellarator and quasi-helical-symmetric (QH) stellarator in comparison with quasi-isodynamic (QI) stellarator (W7-X) and axisymmetric ITER tokamak. We find that turbulent transport in QI and QH can be lower than ITER, while QA is much higher than ITER with the same temperature or density gradient. ITG and TEM steady state transport level can deviate significantly from quasi-linear scaling based on the linear growth rate. Further investigation shows zonal flow generation and stability play a crucial role on determining the nonlinear transport level. These results highlight the importance of evaluating turbulent transport for various optimized stellarators and provide insights into further optimization strategies for lower turbulent transport in stellarator design.