RC circuit: C appears to increase in size with frequency
Now at higher frequencies (by lowering the value of the load resistor)
  the capacitor appears to go up in size.


Your IO pin that drives the resistor isn't capable of supplying tens of mA current whilst maintaining an output voltage at Vcc. Basically, it's output resistance is in series with R1 hence, the lower you make R1 the more of a problem it becomes.