Sleep deprivation is making me a terrible parent to my older kid

Our baby is 5 weeks old and barely sleeps. We're doing 2 hour shifts at night and I'm running on fumes. Yesterday I snapped at my 3yo over nothing and he just looked at me with sad eyes. I felt terrible. My wife is breastfeeding so she's tied up constantly and I'm trying to keep things normal for our older son but I'm so tired I can barely function. How long until this gets better? Does the baby eventually sleep longer stretches or am I just in survival mode for months?