This paper presents a novel method for designing reactor-relevant stellarator magnetic fields that combine several key physical properties, including plasma stability, excellent confinement of fast-moving particles, and reduction of turbulence. The method is applied to a class of stellarator configurations called "quasi-isodynamic" (QI), which are known for their ability to confine fast ions and reduce turbulence. The method, termed Stable Quasi-Isodynamic Designs (SQuIDs), is designed to minimize turbulence driven by ion temperature gradient (ITG) and to achieve excellent fast ion confinement, good collisional thermal particle confinement, and small toroidal currents. The SQuID configuration is shown to be MHD stable, with low levels of turbulence, and to have excellent QI quality. The paper also compares the SQuID configuration with another QI configuration, optimized in the same way but without ITG turbulence minimization, and finds that the SQuID addresses all the criteria set out to achieve. The SQuID configuration is shown to have a lower ITG-driven turbulence than the W7-X stellarator, and to have better performance in terms of fast ion confinement and neoclassical transport. The paper also discusses the importance of the max-J property in achieving good confinement and stability, and the role of zonal flows in reducing turbulence. The results show that the SQuID configuration has good ideal MHD stability properties and is more stable than the W7-X stellarator. The paper concludes that the SQuID configuration is a promising candidate for reactor-relevant stellarators.This paper presents a novel method for designing reactor-relevant stellarator magnetic fields that combine several key physical properties, including plasma stability, excellent confinement of fast-moving particles, and reduction of turbulence. The method is applied to a class of stellarator configurations called "quasi-isodynamic" (QI), which are known for their ability to confine fast ions and reduce turbulence. The method, termed Stable Quasi-Isodynamic Designs (SQuIDs), is designed to minimize turbulence driven by ion temperature gradient (ITG) and to achieve excellent fast ion confinement, good collisional thermal particle confinement, and small toroidal currents. The SQuID configuration is shown to be MHD stable, with low levels of turbulence, and to have excellent QI quality. The paper also compares the SQuID configuration with another QI configuration, optimized in the same way but without ITG turbulence minimization, and finds that the SQuID addresses all the criteria set out to achieve. The SQuID configuration is shown to have a lower ITG-driven turbulence than the W7-X stellarator, and to have better performance in terms of fast ion confinement and neoclassical transport. The paper also discusses the importance of the max-J property in achieving good confinement and stability, and the role of zonal flows in reducing turbulence. The results show that the SQuID configuration has good ideal MHD stability properties and is more stable than the W7-X stellarator. The paper concludes that the SQuID configuration is a promising candidate for reactor-relevant stellarators.