People have a tendency nowadays to assume that whoever had the first or biggest (in their memory) hit recording of a song wrote it.
(I don't think that was the case much before the 1960s - the era of "singer-songwriters - though.   I'm discovering now that quite a few popular "singers" and "bandleaders" from the 1920s thru 1950s were also composers or lyricists on some of their hit songs too, but they are seldom described as such.)

Also, when people do credit the "songwriter" they tend to credit the composer of the music. Lyricists are given short shrift or forgotten far more often.

E.g., "Elton John songs" usually have lyrics written by Bernie Taupin.
Carole King often gets all the credit for songs she co-wrote with Jerry Goffin (that is, when James Taylor isn't credited instead of either of them).
"Andrew Lloyd-Webber songs' usually have lyrics written by Tim Rice.
George Gershwin is very often given full credit for a song when Ira or Irving Caesar or DuBose Heyward, etc., wrote the lyrics.
A highly-acclaimed lyricist such as Johnny Mercer may be the exception. He seldom wrote music, even for songs that have become known as "Johnny Mercer songs."
Then, of course, there's Woody Guthrie, who seldom wrote his own tunes, yet so many of the songs for which he wrote (new) lyrics are known as "Woody Guthrie songs."