Context. The 30 Dor C superbubble is unique for its synchrotron X-ray shell, as well as being the first superbubble to be detected in TeV γ-rays, though which is the dominant TeV emission mechanism, leptonic or hadronic, is still unclear. Aims. We aim to use new Chandra observations of 30 Dor C to resolve the synchrotron shell in unprecedented detail and to estimate the magnetic (B) field in the postshock region, a key discriminator between TeV γ-ray emission mechanisms. Methods. We extracted radial profiles in the 1.5–8 keV range from various sectors around the synchrotron shell and fitted these with a projected and point spread function convolved postshock volumetric emissivity model to determine the filament widths. We then calculated the postshock magnetic field strength from these widths. Results. We find that most of the sectors were well fitted with our postshock model and the determined B-field values were low, all with best fits ≲20 μG. Upper limits on the confidence intervals of three sectors reached ≳30 μG though these were poorly constrained. The generally low B-field values suggests a leptonic-dominated origin for the TeV γ-rays. Our postshock model did not provide adequate fits to two sectors. We find that one sector simply did not provide a clean enough radial profile, while the other could be fitted with a modified postshock model where the projected profile falls off abruptly below ~0.8 times the shell radius, yielding a postshock B-field of 4.8 (3.7–11.8) μG which is again consistent with the leptonic TeV γ-ray mechanism. Alternatively, the observed profiles in these sectors could result from synchrotron enhancements around a shock–cloud interaction as suggested in previous works. Conclusions. The average postshock B-field determined around the X-ray synchrotron shell of 30 Dor C suggests the leptonic scenario as the dominant emission mechanism for the TeV γ-rays.