I had dreadful constipation when I was on thyroxine (synthetic T4.....12 years of it) which resolved itself 6 months ago when I switched to Armour a whole thyroid med containing T1, T2,T3,T4 and calcitonin
Go here
www.stopthethyroidmadness.com
and here for Armour info
www.armourthyroid.com
Do you have any labs and ranges you can post?
You could also try a magnesium tablet daily and extra vitamin c.
You don't mention your thyroid levels, are they in range?
If the thyroid is adequately treated, it's likely not playing a role in the constipation. Would consider a GI consultation.