  Varying the superconducting transition temperature over a large scale of a
cuprate superconductor is a necessary step for identifying the unsettled
mechanism of superconductivity. Chemical doping or element substitution has
been proven to be effective but also brings about lattice disorder. Such
disorder can completely destroy superconductivity even at a fixed doping level.
Pressure has been thought to be the most clean method for tuning
superconductivity. However, pressure-induced increase of disorder was
recognized from recent experiments. By choosing a disordered
Tl$_{2}$Ba$_{2}$CaCu$_{2}$O$_{8+\delta}$ at the optimal doping, we perform
single-crystal x-ray diffraction and magnetic susceptibility measurements at
high pressures. The obtained structural data provides evidence for the robust
feature for the disorder of this material in the pressure range studied. This
feature ensures the pressure effects on superconductivity distinguishable from
the disorder. The derived parabolic-like behavior of the transition temperature
with pressure up to near 30 GPa, having a maximum around 7 GPa, offers a
platform for testing any realistic theoretical models in a nearly constant
disorder environment. Such a behavior can be understood when considering the
carrier concentration and the pairing interaction strength as two pressure
intrinsic variables.
