  The real-space renormalization group (RG) treatment of random
transverse-field Ising spin chains by Fisher ({\it Phys. Rev. B{\bf 51}, 6411
(1995)}) has been extended into the strongly ordered and strongly disordered
Griffiths phases and asymptotically exact results are obtained. In the
non-critical region the asymmetry of the renormalization of the couplings and
the transverse fields is related to a non-linear quantum control parameter,
$\Delta$, which is a natural measure of the distance from the quantum critical
point. $\Delta$, which is found to stay invariant along the RG trajectories and
has been expressed by the initial disorder distributions, stands in the
singularity exponents of different physical quantities (magnetization,
susceptibility, specific heat, etc), which are exactly calculated. In this way
we have observed a weak-universality scenario: the Griffiths-McCoy
singularities does not depend on the form of the disorder, provided the
non-linear quantum control parameter has the same value. The exact scaling
function of the magnetization with a small applied magnetic field is calculated
and the critical point magnetization singularity is determined in a simple,
direct way.
