Estimate of J-20's length is between 20m to 21m. Compare to Su-30's length of 21.94m and Su-34's length of 23.35m, the J-20 is clearly shorter. In addition, J-20's span-to-length ratio is smaller than Su-30. So while J-20 is big, it is not as big as Su-30 or Su-34.
It makes little sense that J-20 being smaller than Su-34 would actually be as heavy. In addition to weight, agility is also dependent on engines, wing loading, and stability of the aircraft.
The more powerful the engines the better, and production of J-20 will employ WS-15s. These will be F-119 class powerplants, not the Al-31F class on the Su-34 which you claimed.
The lower the wing loading the more agile, although now days fuselage produces some lift so we would have to weight in fuselage area some how. In any case, while we have no numbers on the wing loading of J-20 or PAKFA, we do know that F-35 is criticized to have a very high wing loading, and on top of this the aircraft is also overweighted. Based on these information, F-35 would actually be the least maneuverable of all 4th generation fighters.
As for stability, unstable aircraft is more agile than a stable aircraft. Stability measures how easy a system can be disturbed. For example, a normal pendulum is stable because it always go back to its original position, whereas an upside down pendulum is unstable because a slight nudge would cause the pendulum to deviate from its upright position. For aircraft, unstable means a small control input translates to large attitude change. Back in the days when F-15 and Su-27 were designed, flight control system is not sophisticated enough to have unstable aircraft, whereas modern fighter designs are unstable. On top of this, J-20's canard configuration is also less stable than traditional configuration. Su-34 wouldn't come close as far as being unstable is concerned.
So you see, there is little to support the claims that J-20 is not agile and is a fighter-bomber. I will also mention again that J is a designation for fighter aircraft, which means J-20 is not a fighter-bomber.