“Is it normal to cry a lot after having a baby?”

It is called post partum depression. I recommend scheduling some short-term counseling ASAP. A new baby creates major changes for a mother. Sleep deprivation itself has its impact, as does a rearrangement of priorities of time, energy, and responsibility for both parents. These issues should be addressed and not ignored.
 
Marlene N. Kasman, PhD, Licensed Psychologist  

No, it's not normal. It is indicative of postpartum depression. The first thing I would want to find out is whether your mother worked while you were an infant, or if you had to wait for mothering for some reason. If that happened, you would have difficulty nurturing your own child. Your deepest oldest unremembered body memories would remind you that you are giving something which you needed and didn't get. You need to grieve not getting the mothering you needed and find some sort of "revenge" in giving it to your own child. You don't want this to be passed on to another generation. Go into another room, turn up the stereo and cry as much as you can, as often as needed, until you are emptied out of loss. Then come back and behold the miracle. The greatest joy I ever had was in seeing the insight and delight of my child when I engaged him, after I cried away may own pain.

Yes, depression is frequently noted after having a new baby. However, if prolonged or so severe that it interferes with you, family, or care for the baby, please consult your doctor.

Mary J. David ​

In many cases, yes, it is. When you combine the hormonal changes that are occurring within your body after pregnancy and the lack of sleep that usually accompanies the frequent waking by your newborn, your emotional stability is often compromised, and this can result in irritability and, yes,crying frequently.

Check this link out for some insight into your question:
https://rightasrain.uwmedicine.org/mind/mental-health/do-i-have-postpartum-depression-or-just-baby-blues-0