Methimazole is meant to curb your production of T4, so yes, when the dose is too high for you at this point in time, it can cause hypothyroidism. ATD dosing is typically done in a way where you get a BIG dose at first, to bring your levels down quickly and reliably, then they adjust it to see where your "magic" dose is, the one that keeps your levels normal, and stable. ATDs work rather quickly, and that means you can change the dose and feel the effects fairly quickly too. We've seen LOTS of doctors that determine the dose of ATDs by the TSH numbers. At the beginning of treatment, that just seems to create trouble for patients. After you've been on ATDs for a while, TSH is actually a more sensitive and precise number, but at the beginning, as you mention, it lags behind the T4 reality. It's as if the pituitary gland (which secretes TSH in response to the T4 it "senses" in the bloodstream) has gone to sleep in a way, and it takes time for it to wake up and smell the lack of T4, if you will.
It's worth pushing back on the doctor's advice if you are feeling extremely hypo ~ cold, lethargic, things like that. The upside is that you should feel changes fairly rapidly whenever they DO wake up and get the dose adjusted correctly. For now, it'd be good to have a quick conversation with the doctor to see if they're WILLING to go your route, for now, and see how it plays out. By the way, if they do, that's a very good sign for your overall relationship with the doctor. Too many doctors see questions about their advice as some type of assault on their character, and those are the types to run from as quickly as possible.