Her notable roles include Susanna in The Marriage of Figaro, Adina in L'elisir d'amore, Gilda in Rigoletto, Violetta in La traviata, and Nedda in Pagliacci.

[4] She was enrolled in the academy after auditioning with Konstanze's aria "Ach ich liebte" from Mozart's Die Entführung aus dem Serail.

After graduation, Kurzak pursued postgraduate studies at the Hochschule für Musik und Theater Hamburg with Ingrid Kremling till 2004.

By the second year she was assigned roles primarily in German and Italian repertoire, spanning from baroque to contemporary (Thomas Adès' Powder Her Face);[11] and a performance in the 2002 production of Olga Neuwirth's Bählamms Fest.

[19][20] She made other guest appearances at the Bavarian State Opera, in Parma, Toulouse (Gilda in Rigoletto), Salzburg (Ännchen in Der Freischütz), and Palermo (Don Pasquale).

[26] Later in the year, she was featured in two productions at the Theater an der Wien: Don Giovanni (Donna Anna) and Tancredi (Amenaide).

Kurzak sang her first Violetta in La traviata in February 2010 for Mariusz Treliński's new production at the Polish National Opera.

[32] Kurzak was featured in four consecutive seasons in Vienna, where she sang Marie in La fille du régiment and Violetta.

[33] Further debuts took place in Turin,[34] Valencia,[35] Madrid,[36] Barcelona,[37] Los Angeles (Così fan tutte),[38] San Francisco.

[46] In 2015, she performed in La traviata at the Deutsche Oper Berlin,[47] and returned to Covent Garden for a revival of Il turco in Italia.

[49] In 2016, she shared the title role with Diana Damrau in Katie Mitchell's new Lucia di Lammermoor at the Royal Opera, London.

[51] Her 2017 role debuts were Micaëla in Carmen and Alice Ford in Falstaff (Opéra Bastille),[52] Liù in Turandot (Covent Garden), Vitellia in La clemenza di Tito (Palais Garnier).

[53] She was to embody all heroines in The Tales of Hoffmann in Munich, but withdrew due to disputes with the conductor causing her to "lose her passion" for the performance.

[60] She starred in Carmen at the Met, and in summer, Franco Zeffirelli's new La traviata production at the Verona Arena, which was a tribute to the late producer.

[61][62] She made two role debuts: Cio-Cio San in an open-air concert performance of Madama Butterfly at the Parade Square in Warsaw, Elisabetta di Valois in Don Carlo at the Paris Opera.

[68] In February 2015 she took part in Szymanowski's Stabat Mater conducted by Ingo Metzmacher at the Berliner Philharmonie and Wiener Musikverein.

[69] She performed in Nowowiejski's oratorio Quo Vadis on 20 November 2016, and Verdi's Requiem conducted by Jacek Kaspszyk on 27 September 2019, both at the National Philharmonic in Warsaw.