Kind of, but not in the way you’re thinking. The birth canal, as rhubarbarin says, is only a few inches long. For most of labor, the baby’s head is just above that, pushing on the cervix, which is this ring of cartilage that separates the uterus from the vagina (“birth canal”). The pressure of the baby’s head on the cervix may be part of what causes good contractions that open the cervix (we don’t really know entirely what causes contractions, and labor will sometimes start even if they head isn’t pressing on the cervix, but I’m talking generally speaking for a “normal” labor here.)
The cervix is more or less round with what’s usually a teeny tiny hole in the middle, smaller than then a pen. During labor, this hole slowly opens and the edges of the cervix thin out. When the hole is about 10cm across (that’s about 4-5 inches), the head of the baby can fit through it and enter the birth canal. That’s when the mom can start “pushing” to help squoosh the baby down the birth canal. (They probably don’t have to push, actually…unconscious moms can still deliver, because the uterus pushes all by itself. But we’re used to pushing, so we push.) This can take a few minutes or an hour. If it takes more than an hour, most hospitals will make you have a c-section, but this is very controversial. If you’re not in a hospital, it can take longer than an hour and most babies don’t die from it.
Once the head is out, it’s just a matter of seconds to minutes before the rest of the baby is out.
Where the danger comes is if you start seeing specks of black tarry goo coming out before the baby. This is baby proto-poop, and if they breathe that in, it can cause an infection in their lungs. So if they see this stuff, called “meconium”, they’ll stop you from pushing and make you have a c-section. They do not want you to deliver vaginally, 'cause babies tend to open their mouths and breathe in when they’re delivered, and that’s bad if there’s meconium present.
The other danger sign is if the baby’s heart beat slows down while they’re in the birth canal. If it happens at the start of a contraction and the heart rate comes back up pretty quickly, it’s normal and a sign that birth is very very close. It if happens at the peak or end of a contraction, it’s called a “late deceleration” and can be a sign that the placenta might be detaching early or the baby isn’t getting enough oxygen, and it’s worry time.
If the baby is “stuck” in the birth canal, it’s usually a shoulder stuck after the head is delivered, and you’ve got about 5 minutes before brain damage from lack of oxygen. Google shoulder dystocia.