I believe it's important for the children, when they get hurt, to have an adult nearby who takes control, who doesn't panic, and who inspires the child with confidence.
I've seen some absolutely shocking reactions from parents, especially mothers, when a child is injured - hysteria, anger, denial, running away, even attacking... Completely the opposite of what the child needs to see. However, it's not too difficult to correct this situation, to make it more likely that you will act correctly when an emergency presents: the most important thing to realise is that you only panic when you are faced with a situation that you feel you are unable to handle properly. And when the victim is your own child, that feeling is magnified because of your love for that child - a completely natural reaction in normal parents.
So, the most important thing for you to do is to get yourself certificated in First Aid. And then, when something happens, make sure you apply what you learned. It doesn't matter if you REALLY feel overwhelmed and panicky - what the child needs is more important than your personal 'need' to vent/collapse/cry. You need to act calmly, confidently, and positively - but above all, correctly. The course gives you knowledge of what is the best thing to do - so the chances of you panicking, while still a possibility, are much less, because no matter how overwhelmed you feel, you still know what should be done, and you do it.
For the sake of your child(ren)'s safety, all it takes is a little time and application on your part - the courses are not expensive, and actually attending a short course, passing the (rudimentary) test, and thinking things through properly before acting when faced with a medical emergency could very easily save your - or somebody else's - child's life.