Here's another idea, that could be used for a quick check.
One person is located on the causeway ( out of the traffic ) with a uhf-cb radio and a video camera with a good zoom lens mounted 5' above the roadway.
A second person is in a car with a second uhf-cb radio and the laser mounted 5' above the road and pointing back at the observer, the car then drives away from the camera.
When the stationary observer with the zoom lens can no longer see the laser, he calls the car on the radio and the driver notes the distance the car has travelled.
Assuming the video camera and the laser are about 5' above the road surface and the road surface follows the water surface, more or less, then the laser should not be visible when the car is more than about 9-10 km or so away. ( 6 miles )
Allowing for some forward scattering and refraction, you might get a few k's more, but not the full 38 km ( 24 miles )
So if the laser can't be seen after about 9-10 km (6 miles) or more the earth is curved. If the earth is flat ( and the laser powerful enough ) the laser should be visible all the way across the 38km.
You should make sure the observer has the proper laser filter glasses, so as to avoid eye damage if he inadvertently looks at the laser directly.
Be sure to post the video on youtube.
PS Formula used C+R = 0.574*d
2 or if you prefer 5/0.574 = 8.71 = d
2, d (miles) = 2.95 miles to the mid point. This allows for the standard 1/7 for downwards refraction of the sight line.
So with both ends of the sight line 5' off the road surface you should get 6 miles or so. Of course if the earth is flat, you should get all the way across, the full 24 miles.
EDIT: Looking at pictures of the bridge, there is a rise in the roadway, probably for shipping, so you wouldn't actually be able to see all the way even if the earth was flat.
http://www.thecauseway.us/ So you need to work out which side has the longest flat section before the shipping channel and see if it's long enough to do the experiment.