S.H.
She is only a newborn... do NOT do the cry it out "method." She is too young for that. It can be dangerous.
Next, all newborns are like this. Very common... they are used to being all cozy in a womb. Now, they are in a wide open space and having to sleep. AND, they are needing to "bond" with Mommy... very important.
Newborns are also adjusting to "sleep" on "our" terms. Not theirs, like in the womb. So, it's hard for the baby, not only us.
Also in the womb, a fetus will often suck their own fingers for comfort. Sucking is an "instinct" for survival as well.
Next, a baby has to get used to "day" and "night" time and often get confused.
Newborns also have a strong 'startle reflex', meaning they startle awake and their limbs flail. Thus, when put down in a crib, they wake. They cannot control their reflexes yet.. .so it is not controllable.
Some parents 'swaddle' their newborns and this may help.
Or you co-sleep. One alternative, since your own bed is too soft... is for example, (what we did and do), is we have a floor futon in our bedroom, and THIS is where I co-slept with my kids. And once baby is asleep, you can still be with her there or go back to your bed. LOTS of parents do this.
Or, you can try having her sleep in a "moses basket." My son loved his and slept nicely in it, until he grew out of it at about 3 months old.
Next, MANY newborns have 'gas' problems because their digestive systems are not developed yet. It causes them pain and keeps them awake... try infant gas drops for this.
Next, a baby often cannot sleep, because they need more intake. If nursing, make sure she is latched on properly and is actually sucking. Is her mouth completely encircling your areola? Can you actually see her swallowing and hear her swallowing? Is there sufficient milk coming out? Is she nursing long enough to get to the hind milk? This takes about 15 minutes, then they usually nurse longer.
Nurse/feed on demand whether from breast or bottle. Some newborns will need to feed even every hour....
But your baby is NORMAL. And we don't get much sleep once we bring them home. A newborn cannot be expected to sleep all night... or comfortable. They are adjusting to the whole world... to us, to the noise, to the lighting, to smells, to the night time, to gas problems, to nursing, to bonding, to needing comfort, to sleep, etc. etc. It takes a few months, for a baby to adjust to everything. THEN, they get teething and this will be another juncture at which they have sleep fluctuations.
I would try swaddling her. That is what we had to do with my firstborn. She hated the crib. And I co-slept with her. My son was easier. EACH baby is different... so cue into them, and see what THEY need.
I know it's not easy. We all go through lack of sleep, and it's hard seeing our newborn struggling with so much. MOST babies sleep best as we hold them, or co-sleep, or right on Hubby's chest.
All the best,
Susan