We report the detections of molecular hydrogen (H_2_), vibrationally-excited H_2_ (H_2_), and neutral atomic carbon (CI), an efficient tracer of molecular gas, in two new afterglow spectra of GRBs 181020A (z=2.938) and 190114A (z=3.376), observed with X-shooter at the Very Large Telescope (VLT). Both host-galaxy absorption systems are characterized by strong damped Lyman-alpha absorbers (DLAs) and substantial amounts of molecular hydrogen with logN(HI,H_2_)=22.20+/-0.05, 20.40+/-0.04 (GRB 181020A) and logN(HI, H_2_)=22.15+/-0.05, 19.44+/-0.04 (GRB 190114A). The DLA metallicites, depletion levels, and dust extinctions are within the typical regimes probed by GRBs with [Zn/H]=-1.57+/-0.06, [Zn/Fe]=0.67+/-0.03, and AV=0.27+/-0.02mag (GRB 181020A) and [Zn/H]=-1.23+/-0.07, [Zn/Fe]=1.06+/-0.08, and AV=0.36+/-0.02mag (GRB 190114A). In addition, we examine the molecular gas content of all known H2-bearing GRB-DLAs and explore the physical conditions and characteristics required to simultaneously probe CI and H_2_. We confirm that H2 is detected in all CI- and H2-bearing GRB absorption systems, but that these rarer features are not necessarily detected in all GRB H2 absorbers. We find that a large molecular fraction of f_H2_>~10^-3^ is required for CI to be detected. The defining characteristic for H_2_ to be present is less clear, though a large H_2_ column density is an essential factor. We also find that the observed line profiles of the molecular-gas tracers are kinematically "cold", with small velocity offsets of {delta}v<20km/s from the bulk of the neutral absorbing gas. We then derive the H_2_ excitation temperatures of the molecular gas and find that they are relatively low with Tex~=100-300K, however, there could be evidence of warmer components populating the high-J H_2_ levels in GRBs 181020A and 190114A. Finally, we demonstrate that even though the X-shooter GRB afterglow campaign has been successful in recovering several H2-bearing GRB-host absorbers, this sample is still hampered by a significant dust bias excluding the most dust-obscured H_2_ absorbers from identification. CI and H_2_* could open a potential route to identify molecular gas even in low-metallicity or highly dust-obscured bursts, though they are only efficient tracers for the most H2-rich GRB-host absorption systems.

Cone search capability for table J/A+A/629/A131/list (List of fits spectra)