Catalysis of the reaction: a beta-D-galactosyl-(1->4)-N-acetyl-beta-D-glucosaminyl derivative + UDP-N-acetyl-alpha-D-glucosamine = an N-acetyl-beta-D-glucosaminyl-(1->3)-beta-D-galactosyl-(1->4)-N-acetyl-beta-D-glucosaminyl derivative + H+ + UDP.
UDP-GlcNAc:Galbeta-(1,4)-GlcNAcbeta-r-beta-(1,3)-N-acetylglucosaminyltransferase activity