Where would it be cheaper for an American with US dollars to travel for a vacation, Turkey or Brazil?

Brazil. place is good but there are lots of thieves there...so be careful.

Flying to Brazil would be cheaper certainly. A vacation in Turkey, however, would be less costly in terms of hotel prices and other expenses in Turkey. US dollar is exchanged at a high rate at one US$ is equal to 1,23 Turkish Liras.

Visiting Turkey would also be a better experience. You should visit Istanbul, Antalya or Cappadocia if you're interested in history, architecture, culture in general. If you're thinking of a sun-sand-sea vacation, the prices would be even cheaper at the Agean coast.

Brazil

Someone who has been to both countries can advise you. Cheap is easy to find, but cheap AND good?

I don't know. I think you should read comments on touristic web sites very carefully.

Brazil could end up being cheaper since it is much closer. However, also keep in mind that visa requirements for Brazil for Americans are much harder and more expensive, since their immigration policy is "treat the way you are treated".

Brazil will also certainly be much more dangerous.

However, aside from these facts, I think how much you would enjoy either one depends on what you are looking for. From my experience, Brazil has a more active culture that is all about beaches (at least in Rio), working out, going out to bars, clubs, and hanging out with friends. Naturally scenery is incredibly beautiful but architecture, historical monuments or the physical city is not the most exciting.

Turkey, on the other hand, has a lot more history going on. And while people in Turkey also love hanging out with friends, it is more social (conversations over long dinners, backgammon, tea) than mere clubbing (which, of course, also happens).

So, if you are looking for more of a "spring break" type of vacation, go to Brazil (and be very very careful with your bags, camera, jewelry). If you want more of a cultural trip, go to Turkey.