3 week old baby staying up for long periods of time. Is this normal?

My 3 week old baby started off the first week just sleeping and waking up to eat and going right back to sleep. She would wake up 2x during the night to be fed and was able to sleep for about 3 hours straight. The past 2 weeks her night sleeping has not changed though she eats much more now. When she gets up around 8-9 a.m. I stay up for the rest of the day and she is fed, burped and changed. Nothing will put her back to sleep though and she is up for sometimes 5 hours straight usually it's at least 4 hours. During this time I will give her a bath, feed her if she seems hungry, put her on my chest, rock her, bounce her, you name it and usually she will doze off a little when sleeping on me but as soon as a put her down she will sleep for about 10 minutes then wake up crying. During this time period I can get nothing done and have to literally just lay in bed trying to get her to fall back asleep. After about 4-5 hours she will finally fall asleep and sometimes sleep close to 4 1/2 hours and at night she is fine. Is it normal for her to be up this long? Any ways to get her to fall asleep after she wakes up in the morning?