Cuprate high-temperature superconductors feature rich phase diagrams due to the presence of various competing degrees of freedom, inevitable disorder and high anisotropy. In cuprates, the value of the upper critical magnetic field (H_c2) and the role of charge and spin orders have long been under debate. In order to reveal the nature of the quantum phases (T -> 0) and transitions between them, in a varied magnetic field, as well as to investigate the precise interplay of charge order, disorder, and high temperature superconductivity, we perform transport measurements on (1) underdoped Bi₂Sr_0.16La_0.84CuO_6+δ, which exhibits short-range charge order, (2) overdoped Bi_2.1Sr_1.9CuO_6+δ and (3) highly overdoped Bi₂Sr₂O_6+δ, where charge order has not been observed yet. Measurements of linear transport, non-linear transport and Hall effect were performed in magnetic fields (H) up to 45 T and temperatures (T) down to 17 mK.