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Yes, it's almost always a suspension issue, especially if the car bounces on gentle curves too. Check the condition of your shock absorbers (they might be old or worn out) and springs, then see if the stabilizer bars are worn or if you have any alignment problems. If the car only bounces in fast curves, it could also be uneven tire wear or misaligned settings, so before you spend money it's worth doing a full wheel alignment check.
If your car is bouncing and swaying especially in fast corners, before you look at the shock absorbers check the overall suspension setup: often it's a matter of loose or worn anti-roll bars that make the body lean and create instability.
The shock absorbers obviously matter, but if they're old or blown you usually feel more of a continuous "bouncy" effect, whereas fast corners tell you something different. Anti-roll bars (those rods that connect the wheels on each side) are what keeps the car stable in corners: if they're loose, maybe a rubber bushing gave way, you feel it bouncing all over the place. Check the connections with a wrench, push the bar and see if it moves more than it should.
Then yeah, take a look at the springs and shock absorbers too, but start there. If you're unsure, go to someone who can put the car on a lift and do dynamic checks - that way you see right away what's giving and what's not without wasting time.
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