The corticoid effect on TSH is one of those non-linear processes in corticoid pharmacology. The data, and one of the best papers on this is a 2009 review [PMC2784889] shows that there is a dose at which TSH can be lowered – BUT there are a few mitigating factors:
1- the use of physiologic hydrocortisone doses (up to 35-40 mg of oral hydrocortisone) actually support NORMAL thyroid / TSH function.
2- very potent corticoids (Decadron) can suppress TSH but the listed dose of Prednisone to suppress TSH is estimated at 30 mg (or 120 mg of hydrocortisone equivalents).
3- There is a functional phenomenon that causes a collateral “support” need between Adrenal and Thyroid function so that when one is supported and the other is not you can cause lowered function in the one not treated. Ergo, adrenal Tx can help then stress thyroid function and vice-versa.
So all in all, the typical use of escalating and de-escalating cortisol doses in the physiologic range only normalize TSH and thyroid function which actually stabilize the thyroid Tx over time. Only very high doses of stronger steroids with more specific glucocorticoid activity (recall hydrocortisone is much weaker than all other steroids as it has mineralcorticoid activity at a higher level) would ever cause much effect on TSH.