If he'd put you down to half, you may notice improvements, but half of what dose? I found 25mcg a little overwhelming and started on 12.5, moving slowly (and as quickly as I could) up to 24mcg... i.e. after an initial 2-3 weeks of becoming comfortable, I was to alternate with the higher dose and so on.
It was quite difficult at first knowing which symptom belonged to my 'hypo' and which symptom came from the increase in meds, but it didn't take too long to understand it better (with help from this forum).
I agree with gimel in that you'll get great advice here if you can specify a bit more re symptoms, blood tests results and dosages.
(I was given sleep aides to help me with the roller coaster of adjustment period which have enabled me to keep working - I don't need them all the time, but it helps)
Rather than side effects from Synthroid, hypo patients are much more likely to be improperly tested and treated, resulting in levels of Free T3 and Free T4 that are too low in their ranges, with resultant hypo symptoms.
What symptoms do you have? What is your daily dosage? Was the cause of your being hypo identified?
If you will post your thyroid related test results and their reference ranges, members will be glad to assess the adequacy of your testing and treatment.