Welcome to the board although I'm sorry for the reason you're here.
Several things in your post caught my attention for one, I don't agree with your doctors that your levels are oh so normal (aka within ranges) so your thyroid would be fine and would not be causing symptoms.
1) TSH can be influenced by several factors (fluctuates up to 3 points during the day - what time has your levels been done); point is it can give an indication of hypoT but it does not give sufficient information
2) the reference ranges of TSH are too broad, healthy people will have a TSH of around 1 (or little above); AACE has stated that TSH of >2.5 is suspicious of hypoT.
All this just to say your levels aren't all that great, so it's good that your doctor started treatment (because imo you need it).
Were you ever tested for antibodies? might be good to have those done too so you know whether you've got those or not.
Symptomwise: TSH does not cause symptoms, it's not an actual thyroid hormone (FT4 and FT3) are the actual thyroid hormones and the ones causing symptoms when they drop too low for the body's need. Healthy people will have FT4 and FT3 about 60-80% of the ranges, so keeping that in mind, near bottom might be within ranges but sure can be (and will be for most) causing symptoms.
With those levels I'd be feelign very sick so again so yes even with levels looking a bit off at first sight one can feel very sick.
within ranges is not good enough when it concerns thyroid.
As for the timing: think 2 things are to keep in mind
1) thyroid symptoms can come and go, migraines for a few days, gone again and it's like that for a lot of symptoms, so the medication of today is not solving the symptoms of today, does not work like aspirin.
2) there are adjusting 'symptoms' or an adjusting phase: meaning after every dosage adjustment/increase the body needs time to adjust (in general that starts about a week later).
In general it boils down till :some don't feel much, others experience hypersymptoms (those flare out and are temporary) and others feel improvement soon since the body gets what it needs and like it.
all this is temporary until levels are at a good point, stabilized, optimalized. so it's possible that there is improvement in energy level, it's also possible that this seems to worsen again (the dosage still needs to go up), but all that is part of the meds titrating process.
In short: it does take time for the meds to build up in the blood, the meds titrating process takes time and achieving wellness again requires patience, but it's possible that one feels some improvement rather soon, it's also possible that that improvement does stick yet, all that is quite individual.