If I brine my fresh turkey on Sunday night I won't be cooking it until Thursday. Do I have to worry about it spoiling or anything like that or does the salt slow down that process?
I would say it's probably fine. It really depends on how long the turkey has been thawed and how much salt is in the brine.
A lot of people these days say that you should use your meat within three days of thawing it. Actually, there are a lot of other factors that will determine how long you can keep meat before it spoils, but as a general guideline, it's not a bad one. So, if you are putting the turkey in the brine after (completely) thawing it, then three days is well within range.
So long as there is enough salt, the brine won't increase the rate the meat spoils, in fact it will extend the shelf/fridge life of the turkey. I'll brine something up to a week, depending on what I need it to do, and the composition of the brine. Historically, people would keep meat in brine as a way of preserving it for months, even at much warmer temperatures than a typical fridge. Though, they knew what they were doing and the brine would have been a lot saltier than most recipes used today. The book Charcuterie has a great overview of brining (both short and long term) and covers the current food safety standards. http://www.amazon.com/Charcuterie-Craft-Salting-Smoking-Curing/dp/0393058298
Thanks so much trampledbygeese. I'll be using Russ Parsons dry brine on a 10lb turkey so about 2 tablespoons of salt total according to the recipe. This has been bugging me all week! Your answer has eased my worries.