The problem would most likely pertain to your CPU or RAM than your video card. You say you can run Battlefield 3 on max settings, a highly graphic intensive game compared to simple Civ 5, so the graphics card is ruled out for the moment.
Civ 5, unlike Battlefield 3, is very, very CPU and RAM intensive when incrementing the turns, because of the myriad of events being calculated all at once. I have rarely heard of players playing with as much as 50 cities, but I do not see it as something that should be impossible.
Go into the Event Viewer under "Administrative tools" in the Control Panel, expand the "Windows logs" folder and click "Application". Look for an error entry (red cross) pertaining Civilization V there, and see what the entry says caused the crash.
If you fear there might be a hardware problem with your computer, you can try using Memtest86+ to check your RAM, and Prime95 to test your CPU. Memtest86+ is best used from a bootable drive, whether it be a flash stick or a simple CD, while Prime95 is a simple application you can run while your system is active.
Memtest86+: If you have a flash stick/USB pen drive, you can use the auto-installer from their download page to install Memtest86+ on a pen drive. After having done that, restart your computer with the stick connected and go into the multi-boot menu. You do this by pressing a button during startup, but which button depends on your computer make. It's F12 for Lenovo and Fujitsu computers, and F9 for HP, just to give you some examples. Select your pen drive in the menu, and after a few second a blue screen with Memtest86+ should pop up and start testing your RAM immediately. You can see the progress in the top-right corner. If Memtest86+ finds errors on your RAM sticks, that stick is garbage material. If you've got warranty on the machine, you can apply to your retailer about the matter. If you don't have warranty, RAM is very cheap to buy yourself nowadays. Just find the right type and feasible sizes of the stuff, and order them online.
Prime95: Prime95 is so darn simple you can just run it and give it some time to crash your computer should there be a problem with your CPU. You can download Prime95 for 64-bit systems here. Prime95 does what you'd expect from its name: it calculates performs an enormous amount of calculations at a time, fully stressing your computer to the max. If something, whether it be your computer as a whole or just Prime95, crashes during Prime95's testing, you have a rather valid sign that there's something wrong with your CPU. If your computer abruptly shuts down while running Prime95, it just overheated. Prime95 is very intensive, and used to torture test overclocked computers for their stability. Your computer is not likely to take harm from Prime95 if it's from after about 2003.