The shape of the turbulent energy spectrum in the dissipation range, where viscous effects dominate, remains an open question despite decades of work. We report an experimental investigation of intermediate dissipation range energy spectra in turbulent shear layers at Taylor-scale Reynolds numbers, $R{e}_{\lambda }$, ranging from approximately 450 to 1500, which are among the highest achieved in shear flow experiments that resolved small scales. We generated turbulent shear layers in a wind tunnel and measured using nanoscale hot-wire probes with a sensing length $l_w\approx (0.2-0.5)\eta$ that was smaller than the Kolmogorov scale $\eta$ at all $R{e}_{\lambda }$. The measurements resolved wavenumbers up to $k_{max}\eta$ $\approx 17$ at the lowest $R{e}_{\lambda }$ and $k_{max}\eta$ $\approx 1$ at the highest $R{e}_{\lambda }$, where $k_{max}$ is the highest resolved wave number. In the range $0.1\lesssim k\eta \lesssim 0.5$, the spectra collapse onto a universal stretched-exponential form, $E(k) $ exp$(-k)^{} $, with $\gamma \approx 0.5$ independent of $R{e}_{\lambda }$. This value of stretching exponent, $\gamma$, is consistent with recent empirical and computational studies. The Reynolds-number invariance of $\gamma$ is strong evidence for universal scaling in the intermediate dissipation range of high-Reynolds-number shear turbulence.