Don't be confused. Think of it this way. If you are familiar with musicals such as the following (just a handful of examples), you are familiar with many of the most famous songs from The Great American Songbook:
West Side Story
Show Boat
Oklahoma
Carousel
South Pacific
Yankee Doodle Dandy
Meet Me In St. Louis
42nd Street
Hello Dolly
An American In Paris
Singing In The Rain
If you're familiar with any of the songs identifed with people like Bing Crosby, Fred Astaire and Ginger Rogers, Frank Sinatra, Ella Fitzgerald, Dean Martin, Tony Bennett, Judy Garland, Gene Kelly, Doris Day, and others of that generation, you know songs from The Great American Songbook. These works are our cultural heritage as Americans. We should celebrate them.
Most of the “popular music” up until the 40’s or so was from Broadway — that was where American music was made. Even into the Big Band and early Rock & Roll eras, Broadway music still made its way fairly regularly to the top of the “pop” charts.
Nowadays, it would be highly unusual, unless it was a song from a movie conversion or a cover by a top-name artist.