Thanks for posting your message here in the forum. Perhaps some other teachers or students with a similar situation can share their thoughts too. Before I weigh in however, I invite you to look at this picture off Paul Grilley's web site: the lumbar
As you can see, everyone's spine is different. Some people naturally have little space between the spinous processes while others have large gaps. This will ultimately affect how deep you can go in a back bend, once you have worked through all the tensile resistance that at first stops you. In your case, you say your L4 and L5 are "too close". You'll see in the image that for the person on the right, his L1 and L2 are very close also, in fact they are touching. This means that there is no range of motion in a back bend (extension) for that area. But! This is not "bad" - it is simply the way that persons' spine is. Healthy spines may have limited ranges of motion.
In your case, is your L4 and L5 too close because of some injury or other pathology, or is it just the way you are designed?
Yoga will not "fix" the shape of our bones. Yoga can help you open to your optimal range of motion given the bones you have. That is my objective when I do my yoga practice: I seek health, not performance.
So, now to your question: how to strengthen your lower back? You can look at this in two ways: yang exercises to strengthen the core muscles and yin exercises to strengthen the connective tissues.
A resource you may want to check out is Stuart McGill and his books and DVDs, especially one called Low Back Disorders. He explains what happens in my back problems, and illustrates many simple yang exercises. The key is to keep the spine in a neutral position while you exercise and strengthen all the muscles around the core. "All around the core" includes the side stomach muscles as well as the back muscles and not just the stomach muscles, which too many people overdevelop and that leads to low back issues. It is quite easy to program into a regular yoga class positions like the balancing Cat (he calls it the bird dog) and Crocodile and Side planks on elbows (he calls these the side bridge).
For the yin exercises designed to strengthen the connective tissues of the lower back you can do Sphinx, Seal, Straddle (only if it is easy for you), Butterfly, Caterpillar and Straddle (Dragonfly). Twists and Bananasana can also get into some other areas of the lower back missed by the first postures. You can find all these described in the Yin Asana tab on your upper left. There is also a flow for the spine offered here
Good luck with it!