We report magnetotransport studies in thin (<100nm) exfoliated films of CuₓBi₂Se₃ and we detect an unusual electronic transition at low temperatures. Bulk crystals show weak superconductivity with Tc~3.5K and a possible electronic phase transition around 200K. Following exfoliation, superconductivity is supressed and a strongly temperature dependent multi-band conductivity is observed for T<30K. This transition between competing conducting channels may be enhanced due to the presence of electronic ordering, and could be affected by the presence of an effective internal stress due to Cu intercalation. By fitting to the weak antilocalisation conductivity correction at low magnetic fields we confirm that the low temperature regime maintains a quantum phase coherence length Lφ>100nm indicating the presence of topologically protected surface states.