Timing is off.

Put light pressure against the side of the cylinder, and slowly pull the trigger with the pistol empty

The cylinder should lock into the next position before the hammer is release on the trigger pull.
Note, do this for all the chambers in the cylinder.


As for the problem,

Start with measuring the B/C gap between the front of the cylinder and the back of the barrel.
Also note the movement distant that you can pull the cylinder towards the barrel face and than back toward the back of the frame.

As for B/C gap, want something in the .006" range, and in regards to the distance that you can move the cylinder front to back in the framed with the cylinder locked home, want it in the .001" range.

If you have problems here, then the yoke can be stretched, or end shimmed to remove the front/back  slop, and on the B/C gap, barrel can be set back in the frame to correct that problem.

Once these are back in spec, then you need to run the palm pressure test against the cylinder again.  If you still have problems, then could be the hand is work, or the star gears worn on the cylinder.  Also, since the cylinder has to lock before the hammer is release, then hammer, trigger cam could be worn, and releasing the hammer too soon in the cylinder index instead.

Bluntly, since you are asking this question to begin with, sounds like correcting the the pistol is beyond your skill level, and best to just have the pistol rebuilt/ refreshed by a pistol smith instead.

Have him send the pistol to Taurus and they will have it corrected, and back in his hand in less than two weeks.

The cost, just what is costs to ship the gun to Taurus in the first place, since its the gun itself  that has a Taurus Unlimited Lifetime Repair Policy of the gun is self, and not just to the original buyer of the pistol instead.  10th owner of the pistol decacades later,  it's still covered by it life time warranty.

http://www.taurususa.com/repair-policy.cfm