Is it possible to make a turing program into an actual program?
Depending on the version of Turing you have. Some versions of the Turing compiler don't actually compile. That's because the people at holtsoft are too smart. If you have (I think) any version besides Turing 4.0, you should be able to go to Run > Generate stand-alone program.
Like can you post it as a program on a site, say if you click a certain link, you the program runs?
Sure you can upload your program to the internet. As for having the program run when you click on a certain link, that would be up to whether the user chooses to run the program from it's current location or save the program to disk.