This position is fabulous for the care team but so incredibly exhausting for the mother. She is literally pushing uphill. There is no way to get her tailbone out of the way due to it being flush against the bed. Babies are born this way every day, but with a lot of coaching and yelling and counting to ten and purple pushing which is where the mother holds her breath and pushes so long and hard that her face turns purple.
Lamaze International advocates pushing in an upright position to let gravity help you move your baby down and I fully agree. Asking for a squat bar at a hospital should be so that you can get up and squat! Not so that you can tie a sheet to it and pull while you're still laying down with your feet braced on it like stirrups.
Often times, I will see nurses tell a mother she can push how she wants, and then after one or two contractions, say "now let's try it this way" and put her into the turtle position. It is what they are most familiar with and its what is easiest for the careprovider to access the vagina and assess the baby's progress.
As much as I would like to shout, "This isn't working so well, let her get up!" unfortunately that is beyond my scope and would get me kicked out of the birthing room. Moms have to advocate for themselves, by saying, "No, I do not want to give birth laying on my back!" and they have to really mean it. I have also seen a nurse bring a mom who did manage to say that a single pillow, wedge it under one of her hips and say, "There, now you're on your side."
Side-lying is a great gravity neutral position for an exhausted mom who has been laboring for a long time and needs a break, but it is not the best place to start out. Pushing really does work best in an upright stand or squat.
However, a really important piece here is that it does not work to just angle the hospital bed up a bit more if the mother's legs are still out in front of her. By moving to this sitting style pushing position, the mother has changed the shape of her body to an L and has to push that baby around a sharp corner. This is not ideal, although its a great way to slow down a baby who is coming too fast for comfort.
The best position has the mother's body in a straight line, with feet and legs below the pelvis (knees bent in a squat is great) like this | and if that isn't possible, it is better to be completely flat _ than L shaped. Hands and knees can also help a stuck baby, as it gets the tailbone up and out of the way and allows the mom to squat down into the push if she chooses. A disadvantage to the hands and knees position is that it is a lot of work to maintain and some mothers report feeling more disconnected from what is going on and the progress of the baby. It wouldn't work well for a mother who opted to use a mirror to view her baby's birth, for example.
Women can and do have upright pushing take place in a hospital setting, but it is something that they need to be crystal clear is very important to them. The most common situation I have come across is that a mother is laboring as she chooses and feels comfortable and starts to feel pushy. A nurse or doctor tells her that she must be checked in order to begin officially pushing. For the check, she is told to lay down in the bed on her back, and then as soon as it is confirmed that she is complete, she is then given instructions on how to pull back her legs and begin to push. It's seamless really, and many moms never say, "No, I am going to push upright."
For some births, it doesn't matter and the baby makes good progress and comes fairly quickly anyway. But for many women, having the extra room and gravity's help could make all the difference in the world.
More information on the science of why pushing upright can be so helpful can be found here: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1948091/