Why does the current waveform become non-linear in a bridge rectifier?
Current can only flow from the utility (the ac network) to your circuit if the voltage is higher that the voltage on the capacitor.

With no capacitance the output voltage follows the rectified ac and thus the current also follows this (for a resistive load)

As you start adding capacitance to your circuit it is providing "holdup" which creates a flatter, more DC rail. The periods where the AC is higher than your DC become smaller and smaller.

As a result the peak current that needs to flow increases to ensure a consistent amp-seconds to the capacitor.