Make sure that the transition to the animation is instant; and have the animation trigger the line after you have the bullet shoot.
To make sure an animation transition is instant:
-select the transition arrow in the Animator Window;
-in the top right of the inspector there is an icon with a down-arrow next to 3 horizontal lines, click it;
-select Debug from the list that drops down (you can set it back to normal after);
-in 'Transition Duration', change the field to zero.
Also check to see if the animation it's transitioning FROM doesn't have an exit time.
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