There were the large counter-weight trebuchets ( that people are still having fun building and shooting, though no longer to batter down castle walls) and there were smaller traction trebuchets. Instead of a large counterweight, these simply had straps or knotted ropes at the other end of the lever from the sling, and a collection of soldiers on command would grab a line and pull.
Whether they were wheeled or simply picked up by a squad and shifted around is open to question. A reconstruction of a very large trebuchet ( shown as one of the NOVA series) revealed that the wheels were actually rather important for accuracy, at least for the big ones- so there may have been more reasons than portability to mount them.
By the way, trebuchets were the medieval catapults. Torsion catapults, like the onager, seem to have been only a part of the Roman world. They depended upon the ability of fibers ( like twisted hair) to absorb energy and release it, but the fibers available were greatly susceptible to humidity ( many commercial hygrometers actually use horsehair for a sensor ). Not bad for a North African desert, perhaps, but a good English or Normandy rain would put them out of action. If vulcanized rubber or nylon had been invented by 1200, things might have been different.
Research at York University in Toronto in 1989 showed that a small traction trebuchet (powered by twelve well-fed college students) could throw 1.5 kilogram stones around 100 meters four times a minute. W.T.S. Tarver: The Traction Trebuchet Technology and Culture, January 1995
Depends where, for example the chinese loved to use en mass this tiny catapult that was manned by 1 to 3 individuals, the tiny one that was essentially carried and used by 1 person. A more advanced whirlwind catapult
Some more complex variation using more than 2 people like this one.
All are pretty much the same principle, operated by having people, not weights or torsion, but by pulling the rope downwards by squatting and pulling the arms down, using both gravity and muscle power, to move the arm and throw the projectile.
They were light, portable and simple to make, maintain and use, combined with great numbers its rather effective.
