We study the influence of confinement on the rheology and structure of nematic liquid crystals (NLCs). NLCs get confined in networks of titania (TiO2, primary particle size = 21 nm) nanoparticles in suspensions of TiO2 and NLC, N-(4-methoxybenzylidene)-4-butylaniline (MBBA). Suspensions with TiO2 nanoparticle volume fraction (I center dot) of 0.006-0.017, form viscoelastic solids with low elastic modulus (G') of 10(1) Pa-10(2) Pa and short relaxation times. Increase in TiO2 nanoparticle I center dot leads to a rise in G' with TiO2 nanoparticles forming a percolating network at a critical volume fraction (I center dot (c)) = 0.023, and G' of similar to 10(3) Pa. TiO2/MBBA NLC suspensions at and above I center dot (c) = 0.023 show G' similar to omega (x-1) scaling, where omega is the angular frequency and the minimum in loss modulus (G'') with omega. The effective noise temperature, x decreases and approaches 1 with the increase in the TiO2 nanoparticle I center dot from 0.023-0.035, is indicative of an increase in the glassy dynamics. Through the polarized light microscopy and differential scanning calorimetry experiments, we propose that the progressive addition of TiO2 nanoparticles introduces a quenched random disorder (QRD) in the NLC medium which disturbs the nematic order. This results in metastable TiO2/MBBA NLC suspensions in which NLC domains get confined in the network of flocs of TiO2 nanoparticles. We also show that the salient rheological signatures of soft glassy rheology develop only in the presence of NLC MBBA and are absent in the isotropic phase of MBBA.