Topological constraint theory classifies network glasses into three categories, viz., flexible, isostatic, and stressed–rigid, where flexible glasses comprise fewer independent constraints than atomic degrees of freedom and stressed–rigid glasses have more topological constraints than atomic degrees of freedom. For flexible glasses, based on MD simulations of a sodium silicate glass with varying cooling rate (from 0.001 to 100 K/ps), we show that thermal history primarily affects the medium-range order structure, while the short-range order is largely unaffected over the range of cooling rates simulated. This results in a decoupling between the enthalpy and volume relaxation functions, where the enthalpy quickly plateaus as the cooling rate decreases, whereas density exhibits a slower relaxation. We also show that relaxation occurs through the transformation of small silicate rings into larger ones. We demonstrate that this mechanism is driven by the fact that small rings (< 6-membered) are topologically over-constrained and experience some internal stress. At the atomic level, such stress manifests itself by a competition between radial and angular constraints, wherein the weaker bond-bending constraints yield to the stronger bond-stretching ones. For over-constrained glasses, they are expected to exhibit some internal stress due to the competition among the redundant constraints. However, the nature and magnitude of this internal stress remain poorly characterized. Here, based on molecular dynamics simulations of a stressed–rigid sodium silicate glass, we present a new technique allowing us to directly compute the internal stress present within a glass network. We show that the internal stress comprises two main contributions: (i) a residual entropic stress that depends on the cooling rate and (ii) an intrinsic topological stress resulting from the over-constrained nature of the glass. Overall, these results provide a microscopic picture for the structural instability of over-constrained glasses.