Born in Ødis near Kolding, Bjerg attended the Latin School in Kolding before serving an apprenticeship with A.L. Johansen & Son in 1907 during which he created an oak bust of his father. Thereafter he spent an extended period in Copenhagen (1908–11) during which he created a silver medal for a bronze bust of his father. In 1911, he went to Paris to associate with progressive artists of the times such as Picasso, leading to his Cubic bronze bust of the Finnish sculptor Bertil Nilsson (1912).
While in Paris, Bjerg became a member of Section d'Or association, in which Auguste Agero (1880–1945) became a source of Cubic inspiration. With the outbreak of the First World War he returned to Denmark where he crafted Abessinieren (1915), followed by Den svangre (1918), Elskovskampen (1922) and Danaide (1923), of which copies were installed in Copenhagen, Aarhus and Odensen. He later created monuments and statues of other figures which were installed in many Danish towns and cities. From the mid-1920s, he became Denmark's most prominent sculptor creating numerous official monuments in the traditional Danish Neoclassical style.