First you have to consider how diseases are transmitted. STDs are usually skin to skin or warm body fluid to an open sore. Diseases like influenza, ebola, etc can't go through your skin. They require you to touch your eyes, mouth, or an open sore. Even getting a few drops of urine on your legs probably won't get you sick unless you move some of it to a wound, eyes, or mouth. This is why washing your hands is extremely important when using the restroom.
Assuming that the bathroom has regular maintenance and cleaning, the top of the toilet seat is likely one of the cleanest surfaces in the restroom most of the time. The cold hard surface of the toilet seat isn't ideal for any type of disease. Viral and bacterial levels decline very quickly on such surfaces. As long as it is visibly clean there is little to no chance of catching a disease from it. And a disinfecting wipe would reduce the dangers from mild to zero. I'm actually surprised wipes aren't offered in public restrooms. Stores often offer wipes for the shopping cart, but not in the bathrooms for the toilet.
Here is the fun part:
In studies it is actually the handles of the bathroom stall door and sink that are the dirtiest because they are touched after you use the toilet and before you wash your hands.
Secondly many toilets have a violent enough flush to launch a spray of fine mist into the air that no one notices. There is a small chance that could give the flu to people in neighboring stalls. But the person whom uses the toilet five minutes later probably won't have any problems as long as they wash their hands.
Don't worry about STDs. They usually aren't infectious enough or won't survive on cold hard surfaces. Many require skin to skin contact that a toilet doesn't provide. It is usually the more mundane diseases such as influenza or the common cold that are transmitted everywhere that are caught in the restroom.
Truthfully you are more likely to catch something from the shopping cart that you didn't wipe down, than you are the restroom in the same store. That surface is less likely to be clean than the sink in the restroom.
Some reading for the people whom want articles: