Most refrigerators have only one evaporator that handles both compartments. Only the very expensive ones have two evaporators that cool and independently control the temps separately.
Most simply circulate a small percentage of the cold air with a fan in the freezer section (where the thermostat that controls the compressor is getting its signal from) blowing over to the refrigerator section with the hope that that keeps things cool enough in the refrigerator section. If your LG is one of the common ones, it could be something a simple as a large object blocking the air flow between the two sections or a problem with the circulating fan. Also the dial that supposedly controls the temperature in the refrigerator compartment is sometimes connected to a linkage that determines how far a flapper door is help open or closed. Look for a problem in that linkage or damper door.