Huge sympathies, LG&T.I know that drained feeling. It's unbearable watching your child go through something hard.
I have no experience of Turner's syndrome, nor administration of the Grwth Hormone, but i do have a 7 yo currently going through a long procedure which involves lots of painful processes (bone-lengthening - gory details on my blog).
We have found that it really does help to be very mater of fact and upfront about procedures and not try to play them down. DS seems to accept that something will hurt f a short time if he knows it is gouing to stop - and what the mechanism for making it stop is - e.g "the oainkillers will tak efect in about 20 mins" or "physio will be painfu to begin with but each time you do it it will be a little better" "I have 3 more oin sites to clean". His very experienced specialist nurse has a very serious discussion with hi everytime she does something like strut changes and says "how many 'ow's do you think this will take?" and he then yells if it hurts) that many times. i.e she gives hi permission to yell - but also a control mechanism.
Does your DD have any contact with other children undergoing unpleasant procedures? I is of immense help that DS regularly sees other children in clinic having the same procedure, and they chat to each other - quite gruesomely some times, swapping horror stories, but also being very matter of fact 'oh, had that done, it was OK' etc. DS had to have an extra operation and was very very down about it - until he met a slightly older child in clinic wh talked about how many folow-up ops she had - and he then accepted it as normal and not frightening. CAF - Contact A Family can be very helpful puting you in touch with famlies in the same position locally.
We also had a jar, and when we were doing the actual lengthening, we put a £1 coin in each time we turned the bolts - it was good for DS to see that he had a tangible, growing reward.
Also - anticipate your own distress. It is OK to go into the toilet and cry in private - and make sure you have a friend to talk to afterwards. Brownies etc is also very good - immediate distraction and recovery will hopefully increase her confidence with it.
You have every ounce of my sympathy.....and I send a big huge hug.