Heat oil in a large heavy pot over medium-high heat. Add leeks and pepper; cook, stirring often until the vegetables begin to soften, 3 to 4 minutes. Add garlic and cook, stirring constantly, 1 minute more.
Stir in broth, squash, thyme and cumin; cover and bring to a boil. Reduce heat to medium-low and cook until the vegetables are tender, about 10 minutes. dd turkey and corn; return the broth to a simmer. Simmer until the turkey is just cooked through, 3 to 4 minutes. Add lime juice and crushed red pepper. Gently warm the soup until heated through. Season to taste with salt and pepper.