Nicole, I solve that problem by putting the turkey into a large plastic bag. I use two clean white 13-gallon tall kitchen trash bags, tucked one inside the other to give a double strength, although some will recommend to use only food-specific bags. You can also use a LARGE oven roasting bag. Put your turkey in the bag, pour the brine over it till well-covered, press out the air and seal with a twist-tie, set the bag upright down in the cooler, and prop it up with a bag or two of ice and folded towels stuffed in around it. My favorite way of doing a single turkey is to put the bird in the plastic bag, pour the brine in on the bird, and press out all the air possible and seal the bag with a twist-tie. I then place the bag down inside a clean 5-gallon plastic bucket and pack ice around and on top of the bag, and set it in the bathtub (or on the back porch if it's good and cold outside). Refresh the ice every so often to make sure all stays cold.
Just my methods... you do want your entire bird to be submerged, though.
Keri C, smokin' on Tulsa Time