Heat the oil in a large skillet over medium-high heat. Add the red bell pepper and water chestnuts and sauté until softened, about 3 minutes. Add the turkey, garlic, green onions, ginger, and red chili paste and cook, stirring and breaking up the turkey into small crumbles, until no longer pink. Drain any liquid or fat. Add the hoisin, soy sauce, and vinegar, stirring to combine.

Remove the turkey mixture to a serving bowl. Serve alongside lettuce leaves.