Let be an irreducible algebraic curve in ; we are concerned with the maximal order of a torsion point on . We suppose that is defined over a number field , that it is not a translate of an algebraic subgroup by a torsion point, and we denote by its degree and by its genus. It is known that for any , which, as shown below, is nearly best-possible if only the degree is taken into account. Here, by means of a new method, we prove an upper bound (actually for curves in ) which implies in particular . This gives back the above result and for small it improves on it. The appearance of the genus seems to be a new feature in this kind of problems.