We measure the effects of interstellar scattering on average pulse profiles from 13 radio pulsars with simple pulse shapes. We use data from the LOFAR High Band Antennas, at frequencies between 110 and 190~MHz. We apply a forward fitting technique, and simultaneously determine the intrinsic pulse shape, assuming single Gaussian component profiles. We find that the constant $tau$, associated with scattering by a single thin screen, has a power-law dependence on frequency $tau propto u^{-alpha}$, with indices ranging from $alpha = 1.50$ to $4.0$, despite simplest theoretical models predicting $alpha = 4.0$ or $4.4$. Modelling the screen as an isotropic or extremely anisotropic scatterer, we find anisotropic scattering fits lead to larger power-law indices, often in better agreement with theoretically expected values. We compare the scattering models based on the inferred, frequency dependent parameters of the intrinsic pulse, and the resulting correction to the dispersion measure (DM). We highlight the cases in which fits of extreme anisotropic scattering are appealing, while stressing that the data do not strictly favour either model for any of the 13 pulsars. The pulsars show anomalous scattering properties that are consistent with finite scattering screens and/or anisotropy, but these data alone do not provide the means for an unambiguous characterization of the screens. We revisit the empirical $tau$ versus DM relation and consider how our results support a frequency dependence of $alpha$. Very long baseline interferometry, and observations of the scattering and scintillation properties of these sources at higher frequencies, will provide further evidence.
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