This is hard on both child and parent having gone through it many years ago with DD2.
Does your
DD have a special toy. We used to have her "mouse" go with us and it was treated as well as
DD. We also found talking about what was happening when it was happening helped.
To this day she (early 20's) has a thing about needles and usually want either myself or DD1 to go with her when needles are involved.
One thing I learnt was when she had to go to the school dentist was to go with her as she nearly fainted just from fear. Any needles needed at school were missed and given at the Dr's to protect her privacy as kids can be cruel when a child gets upset.
I also used to ring ahead to make sure I didn't have to wait to long in the waiting room. If I had to we would go for a walk or wait outside until it was our turn. Just make sure the nurses know where you are so they can come for you when it's your turn.
If it's a new Dr or nurse pull them aside and quietly explain the situation before hand and this usually makes thing go better.