We have conducted a detailed analysis of scaling for longitudinal and transverse velocity structure functions in a turbulent free shear flow. The free shear flow is generated via a mixing layer under varying conditions of upstream flow disturbances. Two velocity components are simultaneously measured with a pair of cross-wires at two spanwise locations, with varying positions of the second cross-wire, which allows us to study the statistics of two longitudinal and four transverse velocity increments. Spectra, probability density functions of the velocity increments, and scaling exponents are measured and discussed in relation to flow structures such as streamwise and spanwise vortices. Scaling exponents of the velocity structure functions are interpreted in the phenomenological framework of the hierarchical structure (HS) model of She & Leveque (Phys. Rev. Lett. vol. 72, 1994, p. 336). One HS parameter ($\beta$) specifying similarity between weak and strong vortices is shown to be universal for all structure functions, and another HS parameter ($\gamma$) related to the singularity index of the so-called most intermittent structures shows strong dependence on flow structures. The strongest intermittency occurs in the form of streamwise vortices. The results confirm that coherent small-scale flow structures are responsible for intermittency effects and anomalous scaling, and a complete set of measurements of longitudinal and transverse velocity variations are required to derive flow structural information.