Catalyzes the addition of GlcNAc or GlcUA monosaccharides to the nascent hyaluronan polymer. Therefore, it is essential to hyaluronan synthesis a major component of most extracellular matrices that has a structural role in tissues architectures and regulates cell adhesion, migration and differentiation. Also able to catalyze the synthesis of chito-oligosaccharide depending on the substrate (By similarity). [hyaluronan](n) + UDP-N-acetyl-alpha-D-glucosamine = H(+) + N-acetyl-beta-D-glucosaminyl-(1->4)-[hyaluronan](n) + UDP N-acetyl-beta-D-glucosaminyl-(1->4)-[hyaluronan](n) + UDP-alpha-D-glucuronate = [hyaluronan](n+1) + H(+) + UDP Glycan biosynthesis; hyaluronan biosynthesis. Belongs to the NodC/HAS family.