Dealing with a stubborn and challenging little girl

Gaia has a very strong character and when we approach her second birthday, it becomes more and more challenging (even if I was sure it cannot be worse!). She obviously doesn't listen when I say 'don't do it' (apparently they get just an opposite message: DO it but actually how to communicate in a positive way: don't play with doors, don't spill the soup, don't drink water from the fountain???). Tantrums happen from time to time and despite reading some specialists' advice how to deal with them, it is never easy but it is always embarassing in public space. Other people will always judge and assess that you (the parent) does not react properly (you should shout more/less, you should explain more/less, you should allow more/less). Let be blessed the parents of tantrum-free children! But I try to remember it is the phase of her development when she is emotionally vulnerable, she eagers to communicate while she cannot fully yet, she establishes her autonomy and there is basically nothing we can do (apart from keeping her safe when she kicks and is out of control).  
Can it be more difficult? 
(oh yes, my friend, a mum to two teenagers will always say: wait until they are 13...).