Curth signed his first youth contract with Danish club
Farum BK when he turned 16. He was called up for various Danish youth national teams, debuting for the under-16 squad in March 2000. After several training sessions with Dutch club
Feyenoord, he was brought into the youth setup of the club in April 2001. From 2001 to 2003, Curth scored ten goals in 13 games for the Danish under-19 national team, and he was named 2002
Danish Young Player of the Year. He had a hard time forcing his way into the Feyenoord line-up. He was
loaned out to team
Excelsior, based in Rotterdam like Feyenoord, for the 2004–05 season, where he made his senior debut and scored eight goals in 19 games. When his contract with Feyenoord expired in July 2005, he returned to Denmark to play for
AaB in the
Danish Superliga. In his first time with AaB, Curth played both as an
attacking midfielder and forward, before settling as the striking partner of
Sweden international forward
Rade Prica in the
2006–07 Superliga season. On 10. June 2014, Curth signed with
Viborg FF on a free transfer. Curth retired in August 2017. ==Honours==