Yoga poses can be used as leg strengthening exercises in several different ways.
Seated yoga poses with one or both legs straight (and either forward or to the side) can be used to strengthen the hamstrings and glutes.
While bent forwards start with your hands on the floor. Press the heel of the straight leg(s) down into the floor so that the back of your leg activates. So that your leg muscles have more weight to work against lift your hands. Put your hands down and then relax your leg. Repeat a few times, synchronizing with your breath.
To add more weight to the pose so that your glutes and hamstrings have to work harder, reach your hands forwards after lifting them.
Then bring your hands back, put them down, and relax.
Do all actions slowly and smoothly.
Rather than suddenly reaching your hands forwards, slowly reach them. That way you can stop if you need to.
But you also gradually increase the load on your leg reducing the risk of injury.
So that you get better at feeling and controlling your leg muscles, repeat the exercise while synchronizing with your breath. Inhale as you press down and then reach. Exhale as you put your hands down and relax.
One reason for synching your breath with these movements is so that you remember to breathe.
To strengthen the outer hips in bound angle pose, press the feet down into the floor. Again do it repeatedly and focus on slow and smooth activations and relaxations.
In a pose like half split you can strengthen your inner thigh by pressing the foot of the straight knee strongly into the floor.
You can start of with a simple version of this pose, hips over the bent knee, hands on your hips, activate and relax.
After a few repetitions then separate your legs wider with your hands on the floor. Slowly press the foot down into the floor so that your inner thigh activates, and then slowly relax. Repeat.
Another leg strengthening exercise that can work wonders on your glutes is bridge pose.
With hips lifted focus on pressing your feet strongly into the floor to press your hips higher. Keep your hips lifted and then relax your feet. Repeat a few times.
An option is to press down strongly through the shoulders. Then press down strongly through one foot and then lift the other foot.
You can then reach that foot up to the ceiling. Keep exerting a strong downward pressure with your supporting foot and your shoulders.
(You'll probably find that with a leg lifted one shoulder presses down more.)
This next seated leg strengthening exercise I'll call a seated get up.
If you have knee problems either leave this leg exercise out or do it with awareness. This can be stressful on the top knee, so use your hand to help. It can also be stressful on the bottom knee, so be careful particularly when leaning forwards.
I'll break it down into parts here.
Start with one leg in pigeon position (the bottom leg). Swing the other leg over it and place the foot on the floor to the outside of the bottom leg. Place you bottom leg on the floor slightly behind your to help.
Lean forwards inside your top knee, shifting weight to your top leg foot. Use your hand to help. Lift your hips. Lift your hand. Pause and lower. Repeat a few times then switch legs.
Once you are comfortable with this try it without using your hand. Again, be aware of your knees. Desist if you feel they are in danger.
Next repeat the exercise but add a leg lift. Place your hands on your top leg to help support your body weight. Press your hands down strongly against your leg. Then lift your bottom leg. Lower the leg and then lower your butt back to the ground.
Repeat a few times before switching legs or switch legs each time.
As your supporting leg gets more stable you can try it without the help of your hands.
After the last rep you can simply stand up.
One way of thinking about this next exercise is as a way of practicing switching from lunging on one side to the other. (The previous exercise is so that you can work towards standing or sitting just using the strength of your legs.)
Shift weight to one leg, lean your torso forwards as you reach your other leg back behind you. Reach the leg as far back as possible while keeping it lifted, then touch it to lightly to the floor behind you.
Next, bring your torso upright and gently lower your back knee to the floor. Just touch your knee to the floor without relaxing your front leg.
Then lift the back knee. Lean forward and lift the back foot. You can either switch sides at this point or repeat on the same side.
A variation of this exercise is to reach the foot back, keep the foot lifted and just touch the knee to the floor.
A simple action for strengthening the inner and outer thighs is to friction the feet outwards and then inwards. You can do this action in standing side bend, chair pose (keep the knees from moving out or in as you friction your feet) and standing forward bend.
More ways that you can use both friction and pressure to strengthen the legs and arms are included in the video Conscious Muscle Control: Friction and Pressure.
Standing yoga poses like front triangle, triangle twist and triangle can be turned into leg strengthening exercises by deliberately activating either the front of the legs (quadriceps and hip flexors) or the back of the legs (hamstrings and glutes.)
The video Conscious Muscle Control: Quads and Superficial Hip Flexors includes exercises for learning to activate and feel your quadriceps and superficial hip flexors.
The video Conscious Muscle Control: Hamstrings and Glutes includes exercises for learning to activate and feel your hamstrings and gluteus maximus muscles.