One possible source of the problem would be the mechanics of the three primary muscles involved in elbow flexion: the single-joint brachialis and brachioradialis muscles, and the multi-joint biceps brachii. Since both heads of the biceps brachii attach to the scapula—at the coracoid process and supraglenoid tubericle for the short and long head, respectively—the position of the shoulder changes the length of the two muscle groups, and hence the degree and manner of activation involved in the performance of the curl.
We cannot isolate any of the muscle groups, but we can alter their degree and manner of activation by flexing or extending the shoulder (lifting or dropping the elbow relative to the front of the body). Simultaneous flexion of the shoulder and elbow results in a near-isometric contraction of the biceps brachii, theoretically affording it greater strength. This could consequently allow a greater lift while placing a relatively high load on the other muscles in the chain. Swinging the load flexes the shoulder, and this can cause the brachialis to fatigue early. (Note that the medial antebrachial is the nerve, not the muscle.) In order to maximise activation of the biceps brachii, the elbow must be fixed at the side of the body or behind it. Performing curls from a reclined position is one method of achieving this posture, which pre-stretches the biceps brachii.
The other possibility is excessive activation of the carpal flexors: the flexor carpi ulnaris, flexor carpi radialis, and palmaris longus (if it is present). Flexion of the wrist during the curl can works these muscles concentrically rather than isometrically. And since they are smaller than the elbow flexors, they therefore fatigue more easily. Fixing that the wrist straight during the lift ensures that the carpal flexors contract isometrically only, and are hence stronger and more resistant to fatigue.
There is a third possibility, that the muscles of the forearm are, for whatever reason, just weaker relative to the other muscles in the chain. This can happen if your training is unbalanced, or if your occupation otherwise stresses certain muscles more than others. In such cases, your programme may need to be adjusted to limit the overuse of those muscles.
I hope that helps you.