ANYTIME you take in external hormones you will SHUTDOWN any natural hormone production.
Depending on the hormonal system how long it takes for natural production to stop will vary. In the case of the thyroid, typical treatment involves taking a T3, T4 or combination medication.
As the blood levels of these hormones rse the feedback mechanisms in play will tell the thyroid to lower production because it senses there is enough or too much. Generally there is a transient phase where you are getting labs done and altering external med dose as the bodies own production drops. This rollercoaster ride of thyroid hormones can takes months or even years for some people to get properly dialed in.
On the flip side you have supplementation. Its been said that at a 50mg dose of Iodine it can take 6-12 months to become sufficient and likely double that for the body to fully recovery from years of low levels.
Both routs for some offer immediate improvement and for those who just cant take feeling like crap anymore medication is often a faster route.
With that in mind however your thyroid will NEVER function properly while on meds so even if you supplement while on meds you will likely be missing many of the symptoms most folks use to judge the success of supplementation and sooner or later you will need to stop taking the meds and give your thyroid time to start working again to judge the end result of your supplementation.
So going down the medication route may offer more immediate relief of symptoms sooner or later your going to have to pay the piper unless you want to stay on the medication for the rest of your life.