When implementing an animation with turtle, usually the tracer is turned off via turtle.tracer(0, 0) and turtle.update() is called manually.

When the tracer is off, most of drawing functions skips update() calls on the underlying Canvas. But turtle.write() does not do that.
(See the implementation of TurtleScreenBase._write().)

This causes flickering, as can be seen when you run the attached script. (Press space key to hide/show texts.)