"Strong willed" is probably a misleading description. One thing that drew me to Charlotte Mason immediately was her insistence that the child wasn't strong willed--the child had a weak or untrained will.
Better than the term "strong willed" I think is the term "spirited," because that gets more to the heart of the matter. Spirited kids are intense, in many different ways, and their intense tenacity is what gets them the label "strong willed." But just trying to bend their will to yours does not help them to manage their intensity--that intensity is given to them by God and can be an asset, but not until it has been brought under management by their wills. Perhaps they have those intense wills because they will need them to manage their other intensity!
"Raising Your Spirited Child" by Mary Sheedy Kurcinka provides the best insights I have seen into the spirited child. She helps you know whether you have one, what is going on inside of them, and how you can best work with them. Effective strategies for a child who is not spirited will often backfire with a spirited child. Many times, you can tell a child is spirited from infancy. Both of my highly spirited children clearly were spirited from birth--there was no avoiding that reality. Trying to work with them using conventional parenting methods has not worked at all, and anytime I fall back into those methods (often because other parents intentionally or inadvertently make me feel like I'm not doing the right things) we have disaster.
Sometimes people suggest food sensitivities as a contributing factor. Certainly, if you think food sensitivities could be an issue, you should investigate that, but it isn't necessarily an either/or situation. Spirited children often are more sensitive to the world around them, so that they might be more strongly affected than other people by certain foods does not come as a surprise. Avoiding those foods may help tremendously, but if they have other spirited characteristics then other strategies will be needed too.
Parenting a spirited child is exhausting, but your efforts are well worth it. The characteristics that make our job so hard also, when channelled constructively, can be tremendous assets.