Sorry for multiple responses.. but I notice Thyroid Px has high levels of iodine. That is good for someone with a known iodine deficiency, but most people with thyroid problems have an autoimmune condition that's causing it. Taking iodine supplements can actually irritate and worsen the thyroid condition. Did your doctor explain anything like that to you or did the naturopath do any thyroid blood panels to confirm a specific condition?
I should add that some people suspected allergies & problems from fillers and dyes in the pill, but I don't think that's common or even proven. Obviously if that happened you could just try another brand of the drug.
If by "conventional treatment" you are referring to levothyroxine, I don't think you have much to fear. It doesn't have side effects the way most other drugs do since it's a synthetic version of a hormone your body should already be making (but isn't making enough of if you have thyroid disease).
You might have "side effects" if you take too much, but typically people are started on a low dose which is gradually increased. Side effects would be the same as the symptoms for an overactive thyroid. If your dose isn't enough to fully replace the amount of thyroid hormone your body is failing to produce, you will keep having hypothyroid symptoms - not to be confused with side effects of the drug.
I hear some people not wanting to take levothyroxine because they've heard once you start it, "you're on it for life". That's because thyroid disease is usually a lifelong condition and therefore, requires lifelong treatment. If you don't like it, you should be able to stop immediately at any time.
Hope that explains what you're wondering? :)