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 Rotterdam, he was brought into the youth setup of the club in April 2001. From 2001 to 2003, Curth scored 10 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 Dutch team Excelsior Rotterdam 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 Aalborg Boldspilklub (AaB) in the Danish Superliga championship. In his first time with AaB, Curth played both as an attacking midfielder and forward, before settling as the striking partner of Swedish international forward Rade Prica in the 2006–07 Superliga season.