Sink drain pipe leaking
These compression fit plumbing pieces need to fit together very straight to get a good seal.  Try to adjust all the joints to be as straight as possible.  Because each joint can either slide or rotate, this is normally possible to do without much effort.

Also, many bathroom sink drains use 1 1/4" pipe, but many P traps and the pipes coming out of the wall are 1 1/2".  It's a small difference, but it matters to make a proper seal.  Most P-trap kits these days come with an adapter washer that is thicker than the normal 1 1/2" washer.  It may have gotten lost in the packaging, or you may not have gotten one, but they are sold separately and are normally very cheap.