There is a insect, the 4-Lined Plant Bug, that pierces leaves to feed on the sap and these distinctive clusters of damage are its calling card.  It feeds on a couple hundred species of plants during  the 4 weeks of its life span. Eggs are laid in the fall to hatch around this time period. SO, luckily we are only plagued by one generation.  The turtleheads may look a bit ratty for now but will outgrow this damage. You needn't apply a remedy in this case.

Hi, 

Just to follow up, you can visit our Plant Bugs on Flowers page for more information. Plus there is also powdery mildew pictured in the photos which is also very common, especially in the humidity. Both of these issues tend to not severely effect the health of the plants but do make them look a little unsightly. Powdery mildew can make the leaves look a little ratty as well and they can fall off but the plant should be able to still produce flowers and be beneficial to insects. The link provided does mention that you can use horticultural oil to help with the infection, but just be cautious in high temperatures and be sure to read the label. If the turtle heads aren't getting enough sun, perhaps there is a better location for them. 

Let us know if you have further questions.

Emily