You're not doing anything wrong! That kind of nursing pattern, with frequent feedings, and baby wanting to be in close physical contact nearly all the time, is normal, natural, and expected. There are some nursing babies who follow routines that more closely resemble bottlefed babies-- long full feeds every three or four hours-- but they're really not the norm.
Breastfeeding is about so much more than food at this age. Baby has only been in the world such a short time. The world is huge and often overwhelming, and your body and your breasts are home to baby. They are comfort and security and safety, as close as baby can get to being back in the perfect womb environment. There is a complicated hormonal interaction between you and baby that occurs when baby nurses, and it feels wonderful to baby, and promotes sleep. It also serves to keep your prolactin levels high, when baby feeds very often, so that you develop over time a full milk supply.
If you really feel like you need a break, I've had good luck slipping away after nursing mine to sleep when I swaddle them tightly, lie them on their sides, and put rolled up blankets (tape them to make them stay rolled safely) behind baby's back and against baby's belly. Then lie down to nurse, and when you slip away, the blankets help baby to feel like somebody is still there. You have to wait until baby has reach the stage of limp, deep sleep, when you can slide the nipple out easily.
It might be a little early for that, though-- newborns have different sleep patterns. Give it some time!
Learning to nurse with baby in a carrier or sling can make a huge difference. It gets your hands free, so you can carry on your normal activities while baby continues to feel safe, happy, and secure close to your body. Baby can fall asleep in there and sleep close to you, which is what baby wants, but you're not tied to the bed for that time. I would really recommend trying it. It can take patience to learn how to do it, but it's a lifesaver. My kids often spent entire days in there, only coming out to have their diapers changed, in the first five months or so.