I brine large things like turkeys, etc. in a large transparent ziploc bag for clothes storage. A thick garbage bag would work in a pinch.
Put the bag containing the turkey and brine into a larger container (garbage can, crab boil pot, etc.) and fill the larger container with cold water.
This allows you to both get all the air out of the turkey bag before sealing it and then add ice to the water in the larger container to chill the meat without diluting the brine.
Took years and years to figure this out.....but it works great.
Some kind of law of hydraulics at work to squeeze the air out of the Turkey bag.
Also allows you to use much less volume of brine.
No fridge needed.
And, to boot, simply toss out the turkey bag and you are finished cleaning up!
I guarantee this will make your Thanksgiving brining experience far more enjoyable.
You can thank me on Black Friday.