Do a Google search on this site for the problem and you'll come up with I'd guess at least a dozen threads where people of asked the same question, and all got the same answer to solve the problem (and I can't be bothered re-drawing the schematic.)
The root cause of the problem is almost never the loading of the gate capacitance on the op-amp. The gate capacitance doesn't make for a nice response, but it doesn't usually take the opamp into instability on it's own.
Rather the cause is the Drain to Gate capacitance of the mosfet interacting with the inductance of the leads connecting between the mosfet drain and the output capacitance of the DUT (normally a few uH.). This forms a feedback mechanism that quickly gets to zero phase margin as the current increases, and hence the mosfet gain dIds/dVgs increases. If you want to make a load with reasonably high bandwidth, the answer is to use a snubber on the output to dampen this inductance, have a look for Jay_Diddy_B's dynamic load thread, or go download the service manual for the HP 6030 electronic load and have a look.
The fact the LM324 is not rail to rail is of no significance, as long as it can go well below the gate threshold of the chosen mosfet. However, what is significant is the chosen opamp's input range includes the negative rail (unless of course you're using a bi-polar supply.)