A child's ability to self-regulate is the same as the area of the brain they need to use when crossing a street. The prefrontal cortex is an area of the brain that aids with social interactions and the appreciation of other persons. It aids us in settling our intense emotions, generating viable solutions to problems, and making well-considered decisions.
It takes a long time and a lot of practice to develop. The prefrontal cortex is not fully developed until the early to mid thirty in men and the early to mid twenties in women. Nonetheless, we must maintain reasonable, age-appropriate goals in the meanwhile. Children don't learn from harsh words or responses, just as adults don't learn from them. There isn't a single one of us who does. They observe and repeat our actions as a means of learning.
It's important to remember that developing the ability to self-regulate intense emotions takes time, just as it's unrealistic to expect a small child to cross a busy street without help. Our young ones' self-control will improve as their prefrontal brain matures, but it's natural that there will be days when they feel stretched thin, just as it is for all of us. Sometimes, even as adults with a fully formed pre-frontal brain, our strong emotions will cause us to act in ways that we later regret.
Eventually, with the correct experience, kids will develop the ability to self-regulate, just as they will develop the ability to cross the street. In the face of their intense emotions, they need to feel our steady, powerful, and loving presence. Being there for them through an emotional storm is a lot like being a rock in the rapids. Just take a deep breath, empathize, and be present. Stay still until the storm passes. Nothing has to be fixed at this time. They aren't damaged in any way. They need to go through this; it's not a distraction from their development.
Let go of any expectations that they "behave" or that they "control themselves." When we insist on getting our way, we become irritable and frustrated when our plans are disrupted.
Maintaining a close bond is especially important at times of intense emotions. When things quiet down again, you'll have the most impact by doing this. There is no need to discuss what went wrong, how it might be avoided in the future, or what needs to be done to "make things right." They'll be in a better cognitive condition for learning when they're relaxed. Getting this done is not urgent.
Fair standards for ourselves are essential in the same way that developmentally appropriate expectations for our children are crucial. When confronted with profound emotions, we can temporarily lose our ability of reason and revert to acting on gut instinct. Being human means acting in such a way, and it happens to all of us. When we see a child experiencing intense emotions, our natural response is to either fight for them or run away, but we may end up fighting with or running away from the child instead. The leak should be fixed as soon as possible if this occurs. There's a chance to set a good example by showing how to be humble, accept failure, take charge, and make amends, all of which are vital life lessons.