Cut 1 pound turnip into about ½-inch wide sticks (they will look like French fries). Cut 1 small beet into about ½-inch cubes.
Toss the turnip sticks with 1 teaspoons kosher saltin a medium bowl. Set aside for approximately 1-2 hours, tossing occasionally. During that time, you will notice that the turnips lose some water and become softer.
Add 1 cup white vinegar, 1 cup water, the remaining 3 teaspoons kosher salt, 1 teaspoon granulated sugar, ½ teaspoon whole allspice berries, and ½ teaspoon whole black peppercorns to a small saucepan. Bring to a boil over high heat. Turn off the heat, cover the saucepan, and let the mixture steep for 10 minutes.
In the meantime, rinse the salt from the turnip sticks. Pack them into a clean 1-quart jar with the cubed beets and crushed garlic.
Pour the vinegar mixture through a strainer into a bowl or measuring cup. Discard the spices and pour the vinegar mixture back into the saucepan. Bring to a boil again over high heat, then carefully pour the hot vinegar into the jar to cover the turnips.
Let the jar cool to room temperature, then put the lid on and put the jar in the fridge. Refrigerate for at least 2-3 days before serving, or up to a month. Serve with shawarma, donair, or falafel wraps or salads, or enjoy them on their own.