Yes, this is pretty much "amateur hour". The first thing you do when putting panels side-by-side is drill holes in the stud between so you can pass a few conduits between the panels. That makes it easy to relocate circuits, you can just pass the hot and neutral through the pass-thru, and don't need to re-route the Romex cable. The feeder wires need to all be in the same conduit. And this also makes the feeder wires MUUUUCH shorter, which makes them much cheaper and pays for all the conduit bits and then some.
#4 copper is good for 85A.
#4 aluminum is good for 65A.
The feeder needs to be protected by the appropriate sized breaker (rounding up to the next size is allowed, so 70A or 90A breaker). Tapping the utility side of the main was illegal for 2 reasons: that and also the terminals are not certified for 2 wires, especially not 1 copper and 1 aluminum wire. Nothing wrong with aluminum as long as you respect terminal rules. (in the 1970s, they didn't).
You will need to identify either one 240V circuit or two 120V circuits in the main panel, and relocate them to the subpanel via a pass-thru. It looks like any of the circuits on the left side could be re-routed without needing to extend their wires.
Once you have 2 circuits freed, you can install a 70A or 90A breaker in the main panel to feed the sub properly.
Removing the wires from the main breaker will require a "meter pull" because that is service wire that is energized 24x7. I'm guessing the last guy just loosened the lug and slid the wire in there. The meter pull may in turn require a permit from the city.
Note how the drywall is proud of the service panel face, so the panel cover does not screw all the way down. You're accustomed to seeing this with junction boxes but it is not allowed with service panels.
So you have a bit of a task list here.
- Have meter pan pull, and disconnect illegal subpanel wires from main breaker.
- Optional: During meter pan pull, reposition main panel so it will be flush or proud of properly installed drywall (this is not).
- Disconnect and remove subpanel to allow drilling holes through studs for pass-thru's. Preferably 1 large and two 3/4" pass-throughs. The large is for the feeder. Choose places where the knockouts on the two panels line up. Should be doable - they're both GE.
- Reinstall subpanel and install RMC conduit nipples.
- Purchase accessory ground bars for subpanel (model numbers listed on label). Move grounds to them. Remove subpanel neutral-ground bond.
- Replace feeder with a legal 4-wire feeder bringing neutral and ground separately. Choose wire size appropriate to the ampacity you want. #1AL or #3Cu for 100A. #2AL for 90A (significantly cheaper).
- Re-route hot and neutral of 2 circuits to new panel to breaker of appropriate size.
- Position feeder breaker so it is opposite SMALL breakers, to avoid overloading bus stabs. (prior nitwit needlessly put 110A on top 2 bus stabs).
- Install EV charging circuit or better, empty 1" conduit into garage. I know it's needless, but you'll get significantly better offers when you sell the house. Shame not to do it while you're in here.
- Drywall to suit.