Electronic load, Problem turning MOSFET on
The LM324 is old. Without too much analysis...

It could be that the voltage offset (which for the LM324 would be high at 3mV) is causing problems with the feedback loop as there would be unertanty when the voltage across R6 goes lower than roughly 3mV (the input offset voltage will vary from amp to amp and temperature). Another problem could be the common mode range. Anyway this is an old opamp, my suggestion would be to find a new rail to rail opamp with a low input voltage offset.