The structure of free-surface flows in a straight compound channel was investigated in a laboratory flume, consisting of a central smooth-bed main channel (MC) and two adjacent rough-surface floodplains (FPs). The experiments covered both uniform and non-uniform flow conditions, with the latter generated by imposing an imbalance in the discharge distribution between MC and FPs at the flume entrance. The non-uniform cases involved transverse currents directed from MC to FPs and vice-versa. The focus of the study was on assessing the effects of transverse currents on: (i) transverse shear layer and horizontal Kelvin-Helmholtz type coherent structures (KHCSs) forming at the interfaces between MC and FPs; (ii) helical secondary currents (SCs) developing across the channel due to topography-induced flow heterogeneity; and (iii) turbulent large-and very-large-scale motions (VLSMs). Transverse currents can entirely displace the shear layer over FP or in MC, but they do not alter the KHCSs to the same degree, resulting in a mismatch between shear layer extent and KHCSs length-scales. KHCSs emerge once dimensionless velocity shear exceeds a critical value above which KHCSs length-scales increase with the shear. Three well-established SC cells, which are induced by turbulence anisotropy, are observed in uniform flow and non-uniform flow with transverse currents towards FP. They are replaced by a single cell in the presence of a transverse mean flow towards MC. The spectral signatures of VLSMs are visible at the upstream section of the flume but they quickly disappear along the flow being suppressed by simultaneous development of KHCSs and SCs.