We examine non-Newtonian effects in supercritical flow between independently rotating concentric cylinders of nanostructured self-assembling systems. Instabilities in the flow between rotating concentric cylinders, known as Taylor Couette flow, have been a benchmark for stability analysis. For non-Newtonian fluids, a significant amount of research has been dedicated to identifying the roles of elasticity and shear-thinning using polymer solutions. However, much less is known about the flow stability of self-assembling, lyotropic liquid crystalline systems such as one-dimensional cellulose nanocrystals (1D; CNC) and two-dimensional graphene oxide (2D; GO). Here, we compare how the above systems and their hybrids influence the flow transition en route turbulence. The experiments are performed in ramped inner, outer and counter rotation modes using a custom-built Taylor-Couette (TC) cell based on an Anton Paar MCR702e Space rheometer with twin drives. Meanwhile, in conventional TC experiments, visualization aids are used to visualize the flow transitions. Here, we use a custom TC cross-polarized light setup developed in our group previously. Thus, we map the instability modes through the shear-induced birefringence of the suspensions in the form of colorful flow patterns. Further, the transition sequences are identified using 2D Fourier transform pattern spectral analysis. Interestingly, considering only the case in which the inner cylinder is rotating and the outer cylinder is at rest, CNC suspensions showed Newtonian-like instability modes modified by shear-thinning, as evidenced by higher characteristic wavelengths compared to the Newtonian reference case. However, GO suspensions revealed the onset of axisymmetric toroidal vortices (Taylor vortex flow) having a shorter wavelength than the Newtonian reference and distinct high Re turbulent flow patterns.