Combine turkey carcass, onion, celery, carrots, and bay leaves.
Pour water over ingredients to cover. Bring to a boil over high heat.
Once the stock is boiling, reduce heat to a low simmer. Simmer, uncovered, for 2 hours. (Be sure to keep the stock at a low simmer to prevent too much liquid from evaporating and add more water as needed to keep the ingredients covered.)
Taste stock and add some salt, if needed. (It's a good idea to keep the seasoning to a minimum at this stage. You can always add more salt when you use the stock in a recipe later.)
Pour stock through a fine mesh strainer (or any strainer, lined with cheesecloth).
Cool and store in the refrigerator for up to 1 week or the freezer for up to 6 months.