"I did 10 situps" => (난) 윗몸일으키기(를) 열 번 했어.
"sit-up" is called 윗몸일으키기(your upper body(윗몸) is made upright(일으키다)).
"I did 10 situps" means "I did a situp 10 times", so the counter is "times".
We use the counter 번 or 회 (or sometimes 개) corresponding to "times".
번 is common all around, 회 is a little more formal and not as widely used.
개 is normally for physical things, so it is not right here.
But people often use it for anything as it is the most common counter word in Korean.