  We discuss the damping of strong magnetic turbulence downstream of the
forward shock of young supernova remnants (SNR). We find that strong magnetic
fields, that have been produced by the streaming instability in the upstream
region of the shock, or by other kinetic instabilities at the shock, will be
efficiently reduced, so the region of enhanced magnetic field strength would
typically have a thickness of the order l_d=(10^{16}-10^{17}) cm. The
non-thermal X-ray filaments observed in young SNR are thus likely limited by
the magnetic field and not by the energy losses of the radiating electrons.
Consequently the thickness of the filaments would not be a measure of the
magnetic field strength and claims of efficient cosmic-ray acceleration on
account of a run-away streaming instability appear premature.
