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We report thermal conductivity measurements on single crystals of herbertsmithite, over a wide range of temperatures (0.05-120 K) in magnetic fields up to 15 T. We also report measurements of the thermal Hall effect, found to be vanishingly small. At high temperatures, in the paramagnetic regime, the thermal conductivity has a negligible field dependence. Upon cooling and the development of correlations, the onset of a clear monotonic field dependence below about 20 K signals a new characteristic temperature scale that may reflect the subtle crossover towards the quantum spin liquid regime. Deconfined spinons, if present, are not detected and phonons, as the main carriers of heat, are strongly scattered by the intrinsic spin excitations and the magnetic defects. In view of the colossal fields required to affect the intrinsic spins, most of the field-induced evolution is attributed to the progressive polarization of some magnetic defects. By elaborating a phenomenological model, we extract the magnetization of these main scattering centers which does not resemble the Brillouin function for free spins 1/2, requiring to go beyond the paradigm of isolated paramagnetic spins. Besides, the onset of a nonmonotonic field dependence below about 2 K underlines the existence of another characteristic temperature scale, previously highlighted with other measurements, and sheds new light on the phase diagram of herbertsmithite down to the lowest temperatures.