Heat the oil in a 3-qt. or larger saucepan over medium-high heat. Add the giblets, neck, tail pieces, and onion; sauté until the giblets lose their raw color and the onion softens and begins to brown, about 5 minutes. Reduce the heat to low, cover and cook for 20 minutes; the turkey parts will release a lot of liquid. Add 1 qt. water, bring to a boil, partially cover the pan, reduce the heat to medium low or low, and simmer gently until the broth is flavorful, about another 30 minutes. Strain the broth into a fat separator or a 1-qt. heatproof measuring cup. Let sit until the fat rises to the surface and then pour off or skim the fat from the broth.