I'm not sure if there is a direct correlation between thyroid meds and bg readings. I do know that if I don't take my Levothyroxine daily, I feel crappy and it makes my A1C look worse. I would think it would affect bgs as both are endocrine disorders.
I've been wrong before and am not ashamed to admit when I am.
Duck is very knowledgable on this stuff so hopefully he will chime in and correct me if I am wrong.

My last lab results showed my TSH to be 1.85uIU/mL on 150 MCG of Levothyroxine. The lab that did my results shows reference range for TSH to be 0.36-2.84 uIU/mL. My free T3 is at the lower end and my free T4 is mid.