The USGenWeb Project has been asked not to display the birth records as a list due to privacy reasons. As a result, you will no longer be able to view birth records on this site past the approved census release years. So since the 1940 census was released this year, I will add the 1941 birth records to the site next year and so on.
You can, however, still view them on various websites:
Texas Birth Index, 1903-1997 (Family Search*)
Texas Birth Certificates, 1903-1935 (Family Search*)
Texas Birth Certificates, 1903-1935, Swisher County (Family Search*)
Texas Births and Christenings, 1840-1981 (Family Search*)
* Family Search requires that you create a username for some areas of their website but there is no fee to do so.