Preheat a grill to high. Combine the rice, 12/3 cups water and a pinch of salt in a medium saucepan. Bring to a boil, then reduce the heat to low and stir. Cover and cook until the liquid is absorbed and the rice is tender, about 18 minutes. Let stand, covered, 5 minutes.
Meanwhile, put 1 bell pepper strip and 2 outer rings from 1 onion slice in a blender; set aside. Brush the remaining bell peppers and onion with 2 tablespoons olive oil; season with salt and pepper. Toss the turkey with the remaining 1 tablespoon olive oil and the fajita seasoning; season with salt and pepper.
Grill the peppers and onion, turning halfway through, until tender and charred in spots, about 10 minutes. Remove the vegetables to a large bowl; add the lime juice. Season with salt and pepper and toss. Grill the turkey until marked and cooked through, 2 to 3 minutes per side. Remove to a cutting board; slice into strips.
Puree the reserved bell pepper and onion with 1/4 cup water, the cilantro and 1/4 teaspoon salt. Fluff the rice with a fork and stir in the cilantro puree. Divide the rice among bowls; top with the turkey, vegetables, avocado, sour cream and pico de gallo.