  Properties of water have been well elucidated for temperatures above
$\sim$230 K and yet mysteries remain in the deeply supercooled region. By
performing extensive molecular dynamics simulations on this supercooled region,
we find that structural and dynamical instabilities are hidden in the
experimentally inaccessible region between 235 K and 150 K. We find a hitherto
undiscovered fragmentation from 220 K to 190 K, which is the break-up of large
clusters consisting of molecules with locally distorted tetrahedral structure
into small pieces with one or two isolated defects. The fragmentation leads to
considerable changes in the relaxation dynamics of water. We reveal a crucial
role of specific three-coordinated defects in slow but persistent structural
relaxation. The presence of relaxation due to these specific defects makes
water glass transition temperature $T_{\rm g}$ (= 136 K) extremely low and
explains why the $T_{\rm g}$ of water is $\sim$ 1/2 of the melting temperature
Tm, much lower than the commonly obeyed 2/3 rule of $T_{\rm g}/T_{\rm m}$.
