Air carrier pilot reported the ground crew did a swift push back resulting in tow bar over rotation and aircraft damage.

Narrative

XX is an unusual new gate for us in ZZZ; it's deep inside the international alley way. It also demands the push crew to push and then swing the aircraft 180 degrees in order for the aircraft to taxi out. In my opinion the push crew should pushed the aircraft back first and make a full stop; then the push crew could slowly rotate the nose gear and pull us forward with the 180 degree turn. On the night of Day 0 our push crew did a very swift push without the full stop in between that resulted the towbar over rotate and pull the aircraft nose gear side ways with 3 or 4 jump." One of the nose wheel tire bursted and multiple noses gear pins were sheared off. Afterward the decision was made to pull us into a new gate with a super tug that can scoop up the nose gear but the ground crew manager refused to allow their team to perform the job. The only qualified person was our own Maintenance personnel but he was on the other side of the airport. So we blocked the alley for more than 1 hour. Recommend Push crew needs more training for these unfamiliar new gates in ZZZ"