No, it's normal. It's you man. I'm flying the L-39 right now and I'm perfectly able to fly the jet with no curves at all. It's just a sensitive plane by nature. It's made that way. If you find you can't control it, just practice and you will overcome it.
To explain a bit, what makes an aircraft sensitive is the position of its center of gravity in relation to its center of lift. A plane like a Cessna will have its center of gravity forward and thus won't feel twitchy at all. A military jet or aerobatic plane on the other hand will have their center of gravity more aft, making them a lot more sensitive to elevator input.
Modern jets like the Hornet, Viper, Mirage, etc, have built-in stabilization algorithms and basically help you control the aircraft easily. The L-39 doesn't have that, and that's why it feels very twitchy. But again, it's just a matter of getting used to it.