Why are the controls of transport category aircraft much heavier than light aircraft?
The control forces are there by design.  All airliners have artificial force feel from springs, dampers etc. A B747 flies through the skies a lot faster than a Cessna 172 and bad things could happen if it were easy to deflect the flying controls at M 0.85.

The B737 elevator is a good case in point. It has q-feel (artificial feel) provided by a spring device, and the spring gradients stiffen up as a function of airspeed.  At higher speeds it is quite hard to deflect the column by pushing/pulling and much easier to use the trim button to control the pitch. For Level D simulator control forces we measured the forces necessary to deflect the column fully when trimmed in mid position, which turned out to be a serious body-building exercise:


About 200 N at zero airspeed
About 500 N at 400 knots 20,000 ft. (Measurement taken on ground with a pressure hose attached to the pitot tube, and aircraft on jacks with the landing gear retracted).


By contrast, helicopter flying controls are light because helicopters are inherently unstable and a lot of stick action is necessary, particularly in the hover. In most helicopters the feel forces can be switched off completely with the trim button, resulting in a very light stick, perfectly OK with the relatively docile response of the helicopter.

But a 747 is a stable airframe. A fast aircraft with large control deflections results in high accelerations, which can result in Pilot Induced Oscillations or structural damage to the airframe.



The two measurements of B737 control column, digitised from the original plots and on the same scale: column position on the x-axis, force on the y-axis. By design, force required to deflect the column fully from about mid position is about 40 lbf at zero knots, 100 lbf at 400 knots. This is from an artificial q-feel system and the feel is implemented by design. The vertical distance between the sides of the curve is caused by friction.



At 0 knots:


Notice that around mid the spring gradient is very steep: the breakout, to indicate neutral position.
Then between 0 and 5 inch a spring gradient of 4 lbf/inch.
Beyond 5 inch a relatively flat gradient, to make it not too hard to deflect the column fully.




At 400 knots: it is made hard to fully deflect the column. 100 lbf pull from a seated position is nothing to sneeze at.