Eloğlu moved from Istanbul to Cologne as a four year old. The name "Şiir" is derived from her father, Metin Eloğlu (1927–1985), a painter and lyricist, who was awarded with important literature awards in the 1960s and 1970s.
After graduating from high school in Cologne, Eloğlu studied for four years at the Hochschule für Musik Saar in Saarbrücken. Her first engagements then took her to theatres in Erlangen, Oberhausen, and Cologne.
In 1991, she made her first film with Kinofilm, Happy Birthday, Türke!, directed by Doris Dörrie and acted with Ulrike Folkerts in her first scene. Eloğlu became known to a wider audience between 1993 and 1996 by her role of young assistant doctor Dr. Nesrin Ergün in the RTL series Stadtklinik.
