Normann was soon to achieve a considerable reputation, and his works were acquired by, amongst others, Kaiser Wilhelm II. He also gave lessons in drawing and oil painting and showed great interest in young talent. Amongst other things, he encouraged both Gunnar Berg and Ole Juul to study painting in Germany. Many have also given him credit for discovering Edvard Munch. As a member of the Kunstverein's exhibition committee in Berlin, he was the one who recommended that an exhibition of Munch's paintings should be arranged, something which was to result in what has later become renowned at the "scandalous exhibition".
Norman's works can be found at a number of the major European galleries, but have only been purchased to a modest extent by Norwegian art institutions.