Anxiety in 3 year old

I have a just-turned-3 year-old daughter who, when she gets extremely worked up about something- such as having to go to bed in her own room, or when she starts crying about this, that and everything- gets nosebleeds.  When she sees the blood, she gets even more worked up so they get worse.  I can get her to calm down by having her look into my eyes and distract her with that to get the nosebleed to stop; however, how can I help her to learn how to calm herself down before she gets to that point? The only way I can help her now is to hold her, but I need to start teaching her other ways of dealing with her anxiety.  (She is quite healthy otherwise- we went to the doctor today for her check-up)

[April2]

I've noticed my 4 year old shows anxiety a lot too. Usually it's when we are going somewhere and a few people will start to walk out to the car and he starts getting very upset and saying to wait and don't leave. He will think we're leaving someone or him. He gets all worked up and it's hard to calm him down and explain that no one is being left behind. He also gets upset if his routine is changed. It makes me sad to see him get all anxious like that.
I hope you find some answers soon for your little girl. If you get any good ideas that work let me know and I'll do the same for you. Best wishes.
Peace,
April