Hyperthyroidism can cause high IGF-1 (stable measurement of growth hormone) particularly in women, so perhaps hypo can cause low GH. Thyroid removal should not cause a GH problem. The TSH fluctuation is odd -- have not seen 0.05 to 34.6 in one week -- would double check that -- keeping the thyroid levels stable may help normalize the GH levels. Also ask your endocrinologist to see if other pituitary hormones may be abnormal.
I have autonomic dysfunction which may cause the fluctuation from 0.05 to 34.6 in a week.
Doctors thought it was lab error initially but I was hospitalized shortly afterwards and they repeated the bloodwork - no mistake. I felt like my body was completely out of control and extremely ill. My adrenals have been sluggish as well. All happened after thyroid was removed.