Do the proper names of US bridges have the as part of the name?

If you are asking whether or not the proper names of U.S. bridges include "the" and therefore should be capitalized, the answer is "No."

"The" would be capitalized as the first Word of a sentence - as would any Word that starts a sentence - but not in the middle of a sentence. For example:

• The George Washington Bridge was shut down by the Port Authority.
• The Port Authority shut down the George Washington Bridge.

If you are asking if the definite article "the" is used in front of bridges' names, then the answer would be "Yes". It would just not be capitalized as part of a proper name.