Experiments are described which measured concentration fields in liquid-phase strong transverse jets over the Reynolds-number range $1.0\,{\times}\,10^3\,{\leq}\,Re_j\,{\leq}\,20\,{\times}\,10^3$. Laser-induced-fluorescence measurements were made of the jet-fluid-concentration fields at a jet-to-freestream velocity ratio of $V_r\,{=}\,10$. The concentration-field data for far-field ($x/d_j\,{=}\,50$) slices of the jet show that turbulent mixing in the transverse jet is Reynolds-number dependent over the range investigated, with a scalar-field PDF that evolves with Reynolds number. A growing peak in the PDF, indicating enhanced spatial homogenization of the jet-fluid concentration field, is found with increasing Reynolds number. Comparisons between transverse jets and jets discharging into quiescent reservoirs show that the transverse jet is an efficient mixer in that it entrains more fluid than the ordinary jet, yet is able to effectively mix and homogenize the additional entrained fluid. Analysis of the structure of the scalar field using distributions of scalar increments shows evidence for well-mixed plateaux separated by sharp cliffs in the jet-fluid concentration field, as previously shown in other flows. Furthermore, the scalar field is found to be anisotropic, even at small length scales. Evidence for local anisotropy is seen in the scalar power spectra, scalar microscales, and PDFs of scalar increments in different directions. The scalar-field anisotropy is shown to be correlated to the vortex-induced large-scale strain field of the transverse jet. These experiments add to the existing evidence that the large and small scales of high-Schmidt-number turbulent mixing flows can be linked, with attendant consequences for the universality of small scales of the scalar field for Reynolds numbers up to at least $Re\,{=}\,20\,{\times}\,10^4$.