We have studied the metallicity of 17 high column density forests at $N_{{\rm H}{\sc I}}=10^{15-17}$ cm$^{-2}$ and $2<z<3.5$ toward 7 high-$z$ QSOs taken with the VLT/UVES, in order to study the metallicity of absorption line systems as a function of H I column density and $z$. This relation provides an important constraint on metal enrichment mechanisms of the intergalactic medium as well as early star formation in galaxies. We used higher order Lyman series to measure a reliable H I column density for the saturated Ly$\alpha$ lines, and applied an ionisation correction assuming the Haardt-Madau UV background including both galaxies and QSOs. We have found 3, 6, and 8 systems with no metals, C IV-only, and additional ions other than C IV respectively. A photoionisation model with the Haardt-Madau background generally reproduces the observed metal line ratios. The median [C/H] is $-3.03$ and $-1.96$ for the 6 C IV-only systems and the 8 additional-ion systems respectively. The inferred H I number density is $n_{{\rm H}{\sc I}} \sim 10^{-3.5}$ cm$^{-3}$ and $\sim 10^{-2.5}$ cm$^{-3}$ for the C IV-only systems and the additional-ion systems respectively. When other ions are present, [C/H] of the high-$N_{{\rm H}{\sc I}}$ forest is similar to that of sub-damped Ly$\alpha$ systems.