Combine the rice, 1 3/4 cups water and a pinch of salt in a medium saucepan. Bring to a boil, then reduce the heat to low; cover and cook until the water is absorbed, about 20 minutes. Turn off the heat and set aside, covered.
Meanwhile, preheat the broiler. Peel the carrots and turnip and shave with a vegetable peeler into thin strips. Combine the shaved vegetables with the vinegar, sugar, 1 teaspoon salt and a few grinds of pepper in a large bowl. Mix and massage gently with your hands to evenly coat.
Bring the teriyaki sauce, honey and sesame oil to a simmer in a small skillet over medium heat. Cook, stirring constantly with a rubber spatula, until thickened, 1 to 3 minutes.
Arrange the fish skin-side down on a foil-lined baking sheet; season with salt and pepper. Brush with the teriyaki mixture. Broil until glazed and cooked through, 5 to 7 minutes.
Divide the rice, pickled vegetables and fish among plates. Sprinkle with the sesame seeds. Top the vegetables with the seaweed snack strips.