We investigate the temporal evolution of the geometrical distribution of a passive scalar injected continuously into the far field of a turbulent water jet at a scale d smaller than the local integral scale of the turbulence. The concentration field is studied quantitatively by a laser-induced- fluorescence technique on a plane cut containing the jet axis. Global features such as the scalar dispersion from the source, as well as the fine structure of the scalar field, are analysed. In particular, we define the volume occupied by the regions whose concentration is larger than a given concentration threshold (support of the scalar field) and the surface in which this volume is enclosed (boundary of the support). The volume and surface extents, and their respective fractal dimensions are measured as a function of time t, and the concentration threshold is normalized by the initial concentration Cs/C0 for different injection sizes d. All of these quantities display a clear dependence on t, d and Cs, and their evolutions rescale with the variable ξ = (ut/d)(Cs/C0), the fractal dimension being, in addition, scale dependent. The surface-to-volume ratio and the fractal dimension of both the volume and the surface tend towards unity at large ξ, reflecting the sheet-like structure of the scalar at small scales. These findings suggest an original picture of the kinetics of turbulent mixing.