Sir, I think there are a couple of reasons something like this may be happening. One is since I imagine you’re right handed, the right side of your body is probably the dominant side. It probably takes over a little more in bilateral movements like you said.
So I think some unilateral lower-body movements are especially beneficial to you. Strength imbalances between limbs can eventually lead to injury, and in your case differences in muscle size.
Your “right side” muscles are probably much tighter as well since they’re used to doing so much work. When you lift, take the time to really work on the mind-muscle connection to your left side. Concentrate on making sure your left side is taking on the load too.
Take a look at joint mobility as well. Things like internal/external hip rotation and or uneven hamstring length from left to right may be causing a postural issue when you lift. For example, if your pelvis is being pulled to one side by tight TFL/Rectus Femoris/Hamstrings then you’ll be lifting a little skewed and one side will end up with a greater load.
So take home points are:
-Check left/right flexibility and mobility (hips, hamstrings, spinal erectors, quads, etc, even calves) and work to restore them to equal
-Concentrate on building neural drive to your left side since it’s not your dominant side
Video tape some lifting sessions so you can get a different angle and see if you notice anything unequal.