Heat oil in 6-quart Dutch oven over medium-high. Cook turkey and onion 8 minutes, crumbling finely, to 165°F.
Stir garlic, cumin, chili powder, and oregano 30 seconds. Add chiles, corn, sauce, broth, and salt; boil.
Add shells; steadily simmer uncovered 11 to 14 minutes, stirring bottom frequently.
Add reserved water a few tablespoons at a time if sauce dries before pasta tender. Finish with loose coating sauce.
Remove from heat; stir in pepper Jack until melted. Rest 3 minutes; divide among 5 bowls.