Air carrier Captain reported a Critical Ground Conflict event while taxiing to the gate between another aircraft who was on the ramp apron taxiing at a very high rate of speed. Reporting Captain made an aggressive stop to avoid the other aircraft.

Narrative

I was the PIC and Pilot Flying on flight XXXX from ZZZ to Fort Lauderdale. After a normal touchdown on Runway 28R in Fort Lauderdale and after checking left and right for traffic; we exited the runway at Bravo 3 intersection. The Tower had instructed us to turn left on Bravo and then contact Ramp. We checked left and right for traffic and then turned left onto Bravo Taxiway. The F/O then contacted the Ramp and we were instructed to take Bravo taxiway all the way to T3 and turn right to the gate. The F/O acknowledged and we proceeded toward T3 on Bravo Taxiway. As we approached T3; I checked my side and said clear right" and the First Officer checked his side and said cleared left. I then began my right hand turn towards T3. As the nose of the aircraft passed about 20° towards the turn off; the First Officer exclaimed; stop in a very loud voice. I immediately and aggressively applied the brakes. As soon as I looked up to my shock and surprise; I saw Aircraft Y pass directly in front of us on Tango Taxiway at a very high rate of speed. Once Aircraft Y had passed us; we re-verify we were cleared left and right; and I proceeded to our Gate XX.I believe that when the First Officer checked right just prior to our turning right towards T3 the area was clear. However; because of the Aircraft Y high rate of taxi speed they closed the gap so quickly and luckily my F/O looked out again to clear the area before we made the full turn. The Ramp Controller never gave us any indication that we should follow an aircraft or that there was another aircraft even taxing on Tango. Because of the excellent situational awareness of my F/O we were able to avoid an incident or accident."