I want to brine a turkey for Thanksgiving, using Alton Brown's recipe for brining. My wife is concerned that this will increase the amount of salt and sugar in the turkey, making it unhealthy. How much salt and sugar from a brine would end up in say a 12 pound turkey?
I suggest starting with this Cook's Illustrated brining article for more information on brining.
Much of what would be transferred into your turkey is liquid from the osmosis, but some salt, and sugar if you choose to use it, would be added as well.
If you purchase an "enhanced" turkey that has already been injected, you don't want to brine it. The injection has a similar effect. If you've made an enhanced turkey in the past, you've already eaten a bird with added sodium.
Harold McGee states that the turkey will gain 10% of its weight in a combination of water and sodium. Your brine comes out to about 3.9% salt, so your 12 lb turkey will gain ((12 lbs * .1) * .039) = .0486 lbs = 21 grams of salt from a 3.9% brine. Your average soda has 55 milligrams of salt per serving. The total amount of salt from your brine will be spread over the entire turkey, so divide that amount by the number of servings you expect that turkey to provide to get the number of additional milligrams of sodium per serving of turkey from brining.
If at the end of the day you don't want to brine, in that same article McGee also suggests a way to keep turkey moist without a brine.
EDIT: I did my math wrong above, and have corrected the formula and results.