Screw the Hips!
Yes, screw the hips! I may need a better way to phrase this but if you want lower body stability and therefore strength, you want to screw those hips in.
When we properly screw our hips, our feet and ankle line up along with the knees and we utilise our glutes more. All these factors will make for a more stronger and stable lower body whether you’re doing double leg or single leg movements.
You do it by first gripping the floor with your feet (toes straight or up to 10° turned out) aiming to make a tripod with your feet. You should place weight on your heel, the base of the big toe and the pinky toe, thus creating a tripod. Weight should be evenly distributed amongst all three.
Next you want to create a tightening screwing motion by trying to externally rotate your thigh and feet outwards so your left is rotating counter-clockwise and your right going clockwise. The key is to not actually move your feet when you do this thereby you end up screwing your hips.
The amount of screw is not excessive but just enough to maintain alignment of the ankle, knee (kneecap will be pointing towards the pinky toe) and hip. When you are squatting or hinging the primary driver is still the gluteus maximus driving forward into extension (think of cracking a walnut with your butt cheeks) with a side of hip screw for stability.
Over screwing can cause the ankles to get pushed out too far laterally which is called supination. Under screwing will cause the feet to flatten. We want to screw the hips and feet just the right amount.
This can still be done even on single leg movements. I find it helpful to pre-screw the hips into place before moving your opposite leg into position to create the single leg movement.
Give it a shot and let me know how it goes!