How often do we run with two legs touching the ground at the same time? If we really do, chances are that we are not running. We are walking.
Running requires you to have one leg pushing off the ground while the other leg is reaching forward to cover another distance. We don't push off the ground with both legs.
So, if that's the case. Why do we perform exercises like squats, leg press and leg extensions; that uses both legs, to target our lower body?
Well, indeed exercises like squats and leg presses build strength and muscle size in your legs but they don't provide you with multi directional balance. Ask any heavy squatter to perform 10 reps of single-legged deadlifts and they probably will have trouble with their balance. Why? By training often on machines, the exercise takes away our stabilizers and focuses on the direct prime movers. Most exercises on a machine provide one-directional movement.
What about free-weight squats? Well, for development of lower body strength, this is a good exercise but it does not contribute to single leg strength/balance as much as single leg exercises do. For example, the single leg squat.
The actions of the pelvic stabilizers are different in a single-leg stance than in a double-leg stance. Single-leg exercises force the gluteus medius (somewhere in your butt) to operate as a stabilizer, which is critical in running.
Exercises such as lunges performed with one leg on the bench (a.k.a.Bulgarian Split Squat) is a great single-leg exercise that not only improves on the strength and balance of the lower body, but also the dynamic flexibilty of the hip flexors.
Other than lunges, single-legged deadlifts are recommended as it focuses on your hip extensors, which is mainly your gluteus maximus (Your butt again) as well as increases the strength and flexibility of the hamstrings.
Another challenging exercise is the single-leg squat (a.k.a. Pistol). The single-leg squat develops great strength in the thighs and balance. This exercise is not for the weak.
So what are you waiting for? Get to the gym tomorrow to try out these exercises, and you will have lesser running-related injuries while improving on your running technique.
In summary, single-leg exercises are essential to the improvement of speed, development of balance, and prevention of injury.