It is normal - this is because shes at an age where she wants to be more independent, but can't always do what she wants, and also won't have the language to convey what she wants so shes full of frustration.
BUT it won't necessarily get worse!
My dd had a temper around the same age, and i now have a 2 1/2 yr old angel! Honestly i cannot believe how well she behaves for me now, after all the fighting we had over every little task - getting dressed, putting shoes on etc.
She used to have full on tantrums, if i didn't need to go out, i just left her to a it until she calmed down. If we were in public i'd pick her up, hold her tightly to me (to stop arms and legs flapping) and would whisper in her ear to stop otherwise - (fill gap with an appropriate threat e.g we'll go straight home, you won't get your sweeties, i'll take the toy away etc etc).
The aggressiveness towards parents is horrible, i found it really hard to not react, but you have to avoid attention as much as poss in order for it to stop. I used to say a firm no and then sit her away from me, if she came over to me, i'd say i don't want to talk to you cos you've made me sad and i'd leave her to tantrum about it.
The thing to remember when these tantrums are going on is that they need to let out their frustration and you are not being a bad parent by leaving them. By ignoring, they are learning appropriate behaviour, and they are letting off steam which is far more beneficial to keeping it inside and becoming a withdrawn child who doesn't know how to deal with their anger.
Sorry for waffle, i hope this helps! Just wanted to say with good management it won't necessarily get worse!