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After squat day I have DOMS on my right side only. Therefore, I must disproportionately be using my right side for the lift.
Any suggestion on how to balance my effort better?

Do single leg work part of the time.

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Let thy food be thy medicine, and thy medicine be thy food.~Hippocrates
Strength is the adaptation that leads to all other adaptations that you really care about - Charles Staley
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Thanks TimD

With a goal of hypertrophy my leg day is:
Squat 5x5
leg press 3x15
single leg hamstring curl 3x15
single leg extensions 3x15

The next day I have DOMS in my right glute. Should I drop squats altogether and just I work on single leg presses?

Nah,
Single-leg glute bridges and RDL's work you glutes and PC in general. That could strengthen your glutes so you wouldn't get huge doms for overloading your ass on squats. Don't drop squats. Drop leg extensions. After squats and leg presses they aren't worth that much. Plus it's quite heavy on your knee.

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Prove the impossible!

I am also not a fan of leg extensions. Not a very good exercise, and hard on the knees. My isolation preference is the hack sled machine. So after squats, and leg press that's next. I have to say I am really liking the idea of the belt squats. That looks like a good one. I might need to round up a couple more cinder blocks to stand on, and get one of those belts. I could probably do them in the gym standing on a couple plates too.

I did them in gym on stacked aerobics platforms. When two flat benches were available I also borrowed them. The right belt and weight attachment is very important too. A great exercise.

I'm a fan of leg extensions, I do them at the end of the workout. I can see how they can hurt your knees though. The one I use in my gym is loaded with plates and is at the end of a bench, simple design but never had any knee problems. The machine ones with the weight stacks hurt my knees a bit.

I'm fine with dropping leg extensions, but will that really balance out my squat?

What do you mean by that?
Dropping exercises doesn't necessarily make a difference, adding corrective exercises will change your performance or imbalances.
The leg extensions were just a bit futile after squats and leg presses. Considering that you could do something for your DOMs and other weaknessess. I suspect it ain't the quads that need the extra work.

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Prove the impossible!

Doh! I understand now, add in a corrective exercise to help balance out my squat and drop the redundant leg extensions.

I have a question on this - I get similarly asymmetrical DOMS from bilateral work, and sometimes from unilateral leg work also, but my question for the exercise science types is this: I have read in a lot of places that DOMS is not a good indicator, or an indicator at all necessarily, of how hard your muscles have worked, since there are so many other factors that play into it. Doesn't that also mean that asymmetrical DOMS doesn't necessarily mean that one side has worked harder than the other?

It implies that you have done more that you're used to. So if one side gets more DOMS on any occasion, it implies that what you have done was unaccustomed for that side. So you can assume that that side was a little weaker. Not a precise indicator at all, but it tells you something.

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Our greatest fear should not be of failure, but of succeeding at things in life that don't really matter.--Francis Chan