Theoretically, the increase of Reynolds Number over 10000 would increase the turbulence intensity, If The reynolds Number in your experiment are in the range of 2300 and 10000, try to use a corant, like methylene blue, to visualize qualitatively the disturbance on flow.
The Reynolds Number can be interpreted as the ratio of inercial forces and viscous forces, and this is why it is associated with turbulence of flow. But it measures the average velocity of flow, so, for great velocities of flow, can seem that the turbulence intensity would decrease slightly, i.e., the velocity distribution profile wouldn't look so turbulent because of high fluid velocity profile. On the link below has a description of TI model for turbulent flow on pipes.
Using the equation described on the link, I found a decrease of about 1,55% for Reynolds Number range from 4000 to 51500.
P.S.: I believe this model it's a good correlation only for turbulent flow (Re>2300).
Good job about the TI figure. I am not sure if the turbulence gets fully developed when x/d=8. But your results looks not disagree with what Lucas Queiroz described. Maybe you can repeat your experiment with a longer test section and larger Reynold number.
In fully developed pipe flow at high Reynolds number the turbulence intensity should tend to become constant. Your results show this tendency nicely in the Re=10000, Re=20000 cases.