Instead of focusing on only bending the upper body to the side, focus more on pushing the hips in the opposite direction. If you only bring your upper body down to the side and leave your hips directly over your feet, you are crunching your side instead of creating a nice open C-shape.
Once you reach alignment with your hips square to the mirror, only focus on moving your hips to the side. You want to push your hips so much that they go further out past the sides of your feet. If you only think about bending more at the waist you are not opening up the intervertebral discs. By focusing on moving your hips further out past your feet, you continue to lengthen and open up the body, giving more space between the vertebrae. This will make the opposite side compress more and bring your upper body down naturally. Your arms and hips will pulling in opposite directions.
The “push, and push” at the end of Half Moon is intended to make the hips push out, not just push the body down more. By pushing the hips out, the body automatically has more room to bend while maintaining core strength.
In the photo below, she is mostly trying to bend her upper body to the side, and isn't pushing her hips in the opposite direction. She isn't able to fully lengthen and open up her body. (As a side note, her arms also aren't touching her ears, and her feet aren't together. For more info on alignment in Half Moon, click "here".)
The photo above shows nice alignment with the hips pushing far to his left, further out past his feet, causing a nice open C-curve in his entire body.
(photo from Bikram Yoga Los Angeles)