Essentially, you have out monitiored your video card. Very large monitor area useses much more video horsepower than a smaller monitor. It essentially is almost an exponential progression. Two 24" monitors running in 1920 x 1080 for example would require at least two 460 GTX's in SLI mode to run at medium settings on maximum resolution for most higher end graphics games. Or a single 580 GTX. With one monitor, your one video card's full horsepower is devoted to that screen. With two, you divide that available processing power between both screens. If the new monitors are significantly larger and higher resolution than your old one, which it sounds like they almost certanly are, they require more than your present video card solution can provide.
I'd figure out exactly which make and model of video card you have, and see if I could find a matching second video card (identical lot numbers if possible) and assuming you have a reasonabnly robust power supply and a motherboard that can take it, run two video cards in SLI mode. That or upgrade your primary video card.