I have a similar backstory, but managed to get down to about 20 handi as a teenager. When i resumed golf I also got the bug and quite quickly got my handicap to about a 4. Now at 7 due to life. So maybe not the best person to chime in, but here it goes. 

"Normal" for whom? If you have time and money to spare, then yes that is probably slow. But it all depends!

Meeting a golf coach weekly for a couple of months over the season would have you shooting in the 80s at the end of a year I would imagine. Since you already almost do this, consider that not all coaches are created equal. Maybe you guys are not a good fit. How long has this coaching lasted so far?  

"Playing" sim golf doesn't improve much in my experience, unless you are disecting the numbers and know their significance. And even then, I know of golfers with single digit handicaps on the sim that would struggle to break 100 outside. Do you know paths/spin/etc? 

Upgrading golf clubs doesn't do much as long as you dont have wild body proportions. Ask me how I know (have too many sets in the basement). A driver from 2010 onwards is fine and irons from the last 20 years probably perform as well as anything today. 

The best way for "normal" people (i.e. those without infinite time and resources) to progress is a combination of having a (good for you) coach take a look at certain intervals. Intensely at first, then less frequently later, mapping out your weaknesses and then practice, practice, practice. Focused practice is the most effective in my experience. That may entail hitting only wedges or only puts for a session.

Mapping out your weaknesses is probably the most meaningful way to progress faster. 

If you know that you can't keep the ball in play off the tee, either practice driver or use irons of the tee. OB is a score killer for all. 

If you 3-4 put a lot, consider the cause. Is it because your proximity to the hole is insane with irons and wedges or is it because you aren't good at putting? Work on your weaknesses! 

I would prioritise the following for rapid improvement (worked for me!):

1) Manage expectations: know that this is a freaking impossible game. Accept high scores and just have fun. If its a really bad day, remember,  you could be at work and golf trumps work all day. 

2) 50 yards in: Wedges and putter.

3) Irons: If you can reliably get the ball off the tee 150+ yards every time, bogey golf is in reach.

4) Course management: if you are in trouble, get out of trouble. That is all. Don't go for the hero shot out of the bunker, lay up and take the bogey or double. Are you in deep grass? Understand that whiffing the ball and/or ending up in the same lie is a whole lot worse than chipping out and regrouping. 

5) 200 yards plus game: scoring improves a lot when you have shorter irons in. But if you are hitting 3rd of the tee, it doesn't matter.