The variability of the physical thickness of fully developed turbulent interfaces is examined using scalar measurements in the outer far-field regions of round jets at a Reynolds number of $\Re\,{\sim}\,20\,000$ and Schmidt number of $\Sc\,{\sim}\,2000$. The interfacial thickness is considered in terms of the inverse magnitude of the scalar gradient across the interface and its relation to the scalar dissipation rate. The thickness variations and their conditional statistics are examined on outer interfaces at a resolution of ${\sim}\,1000^3$ with data that capture the full transverse extent of the flow. At the resolution of the present measurements, the interfaces are observed to exhibit highly intermittent thickness variations that consist of striation patterns, or undulations, along the interfacial surfaces. The conditional probability density of the interfacial thickness is found to be nearly lognormal, in agreement with previous studies. A new scale-local density measure of the interfacial thickness is formulated to examine the effects of coarse graining and the dependence of the thickness on resolution scale. The scale-local thickness density, conditionally averaged on the outer interfaces, is found to exhibit self-similarity in a range of resolved scales. This observation of self-similar behaviour, in conjunction with intermittency, provides a physical ingredient useful for studies of phenomena sensitive to turbulent interfaces.