Son won't sleep unless I'm holding him

My 3 week old has decided that the only acceptable sleeping position is on my chest. The second I try to put him down he's screaming. I know people say this is normal but I haven't slept more than 45 minutes straight in like a week. My wife keeps telling me to just let him cry it out but he's so tiny I feel like a monster. Anyone else deal with this? When does it get better?