The breakup of a liquid jet by a surrounding high-speed gas jet with different liquid Reynolds numbers and gas Weber numbers was studied using high speed phase-contrast X-ray imaging technique. Focusing on the liquid core, the portion of liquid that is connected to the nozzle, four distinct morphologies were observed and can be associated with changes in the large-scale configurations of the two-phase flow are reported in a phase diagram. Gas-to-liquid kinetic energy balance arguments capture several transitions between the liquid core regimes. The temporal evolution of the center of mass of the liquid core is extracted to quantify its motion, whose statistics can be utilized as a signature to distinguish different regimes. At low to moderate gas Weber numbers, the dynamics are strongly influenced by flapping, while long-time dynamics develop at high Weber numbers, that give way to quasi-periodic motions when swirl is impeded to the gas jet.