I don’t have one, I just do it at home
It really would be worth going to a few classes for some expert assistance and hands-on correction or adjustment.
Everyone has a curve in their lumbar spine. The idea of Pilates (and some other slow controlled movement techniques such as ballet or yoga or Alexander technique) is that you don't flatten that out, but you do have to work to strengthen the abdominal muscles that hold your centre like a corset.
If you have a pronounced lumbar spine curve, you may need to make an adjustment to the angle of tilt of your pelvis. Or as another poster says, bend your knees a little. But we can't see you, so we can't diagnose or correct what's going on.
It sounds as though you haven't built up core strength yet sufficient to hold the legs in an extended position and protect your back. So you need to see of the programme you're doing has an easier version of this exercise.
Then really focus on small movements, executed really precisely with the correct breathing pattern. Think "Navel to backbone" and try to think of your back and side ribs really spread out on the floor. Ensure there's no transferred tension in your jaw or neck.
You also need to learn to distinguish between 'strain' - the muscles starting to work in ways that they're not used to, and 'pain' - over exerting & damaging your soft tissues.
But unless you're quite experienced & trained in some form of movement and know your own body's capabilities really well, and can feel your skeletal alignment from the inside (as it were), you really can't learn Pilates from a video ...