Oh bless you, those early nights are so tough. It honestly will improve.
Dd1 was like that, it got a bit better by 4 weeks and was much much better by 8 weeks.
Some things to try -
Can you co-sleep and feed lying down? I didn't learn how to do that until dd1 was 6 months. Total game changer with dd2 and saved my sanity!
Getting out for even a short walk in daylight every day will help the baby begin to learn day from night.
Are you breastfeeding or formula feeding? Do you have a partner? If you do, share the load - if breastfeeding, can you express so partner can do one bottle, or do a bottle of formula every night? I used to go to bed about 9pm straight after a feed. DH would hold the baby and do a bottle, and only bring her up if she was either asleep or needed the next feed. Sometimes I got 2 hours sleep, sometimes 4, but it was a lot better than nothing!
If she naps in the day, sleep then. Or ask friends or family to hold her or take her for a walk so you can sleep. It does get easier, but this bit can be really tough, so do ask for help.