Lebesque was born in Nantes, at the Quai Barbin (now dock Barbusse), and had his secondary education in Clemenceau High School. In 1930, he was editor-in-chief at the Loire Echo. Responsible for the Nantes branch of the Breton Autonomist Party (Parti Autonomiste Breton), Lebesque left the latter in 1931 and founded, with Théophile Jeusset, the more extremist movement Breiz da Zont, and its political wing, the Parti Nationaliste Breton Intégral.
During the German occupation of France in World War II, he worked for the collaborationist newspaper L'Heure Bretonne, then for various journals in Paris, where he met and befriended Jean-Paul Sartre and Simone de Beauvoir.
In 1952, Lebesque joined the satirical journal Le Canard Enchaîné, for which he wrote a popular column, modelling his style on Albert Camus, with whom he became a close friend. He eventually became an editor of the journal.
After 1966, Lebesque also participated in the Breton autonomist revue Ar Vro.