Heat oil in a 6-quart Dutch oven over medium-high heat. Add turkey and onion and cook for 8 minutes, breaking the meat into fine crumbles and cooking it to 165°F.
Stir in garlic, cumin, chili powder and oregano for 30 seconds. Add green chiles, corn, tomato sauce, broth and salt and bring to a boil.
Stir in shells, reduce to a steady uncovered simmer and cook for 11 to 14 minutes, stirring frequently from the bottom.
Add reserved water a few tablespoons at a time if the sauce gets dry before the pasta is tender; finish with a loose coating sauce.
Remove from the heat and stir in pepper Jack until melted. Rest for 3 minutes and divide among 5 bowls.