A centre back, Bjelland began his career in his homeland with Fredensborg BI and BSV, before moving into the youth system at Superliga side Lyngby in 2001. Financial problems at the stricken club saw the Royal Blues relegated to the Danmarksserien, but by the time Bjelland made his senior debut in September 2006, they had risen back to the 1. division. Winning the 1.division championship in the 2006–07 season saw Bjelland start his first full season with Lyngby as a first team regular, starting the club's opening 9 games. He dropped out of contention and finished the 2007–08 season with 11 appearances and no goals. Back in the 1.division after relegation from the Superliga, Bjelland was a regular in the 2008–09 season, making 24 appearances and scoring one goal. He made five appearances in the opening month of the 2009–10 season, before departing on 31 August 2009.
Bjelland returned to the Superliga to sign for FC Nordsjælland on 31 August 2009. He quickly established himself in the team and had a successful first season, making 26 appearances and scoring one goal. His campaign was capped by success in the DBU Pokalen, with Nordsjælland beating Midtjylland 2–0 in the final after extra time. Despite missing out on Nordsjælland's short-lived Europa League campaign, Bjelland had another successful season in 2010–11, being named vice-captain and again winning the DBU Pokalen at the expense of Midtjylland.
Bjelland finally made his European debut at the beginning of the 2011–12 season, playing in both legs as Nordsjælland crashed out in the qualifying stages to Sporting Lisbon for the second successive season. Bjelland made 29 appearances and scored one goal in a season which saw Nordsjælland win the Superliga title for the first time in the club's history. He departed the club in June 2012, after making 84 appearances and scoring three goals during three seasons at Farum Park.
On 6 November 2011, it was announced that Bjelland had signed a four-year contract (with an optional further year) with Dutch Eredivisie side FC Twente, effective from the beginning of the 2012–13 season. He had a debut season to forget at De Grolsch Veste, spending six months out with a broken metatarsal and making only 14 appearances. He came back fit for the 2013–14 season and was a virtual ever-present as Twente managed a third-place finish in the Eredivisie. Bjelland was named as captain for the 2014–15 season, though he was in and out of the team, missing matches through suspensions and four separate injuries. He left the club on 2 July 2015, after making 79 appearances and scoring three goals during his three seasons in Enschede.
On 2 July 2015, Bjelland moved to England to sign for Championship side Brentford on a three-year deal for a club-record €3 million fee. The move reunited him with former Nordsjælland coach Flemming Pedersen, then Head of Football Philosophy and Player Development at Griffin Park. A groin injury suffered in pre-season saw Bjelland only fit enough for the bench for the first match of the 2015–16 season versus Ipswich Town and he made his first start for the club three days later in the League Cup first round versus Oxford United. He was at fault for Oxford's first goal of the 4–0 defeat and was withdrawn at half time after suffering a knee ligament injury. He returned to full-contact training in late April 2016 and entered the 2016–17 pre-season fully fit, playing his first match for 11 months versus Boreham Wood on 9 July 2016.
Despite missing part of the 2016–17 pre-season with an abductor strain, Bjelland broke into the starting lineup in an unfamiliar left back role in mid-September 2016. He returned to his natural centre back position after the team's formation was switched to 3–5–2 three months later. Bjelland missed much of March and April 2017 through injury and finished the season with 29 appearances. Bjelland deputised for the absent Harlee Dean early in the 2017–18 season scored his first Brentford goal in a 4–3 defeat to Nottingham Forest 12 August 2017.
Despite being eligible to represent Norway, Bjelland represented Denmark at U16, U18, U19 and U21 level, captaining the latter team and making three appearances at the 2011 European U21 Championship, hosted in Denmark.
Bjelland made his full international debut in a 0–0 friendly draw with the Czech Republic on 17 November 2010, playing the full 90 minutes. He failed to receive a call-up for another 11 months, when he was drafted in for a pair of Euro 2012 qualifiers against Cyprus and Portugal, starting both matches and helping the team to wins. Bjelland scored his first international goal with penalty to seal a 2–0 friendly win over Australia on 2 June 2012. He was called into Denmark's squad for Euro 2012, but failed to make an appearance before the team crashed out in the group stage. Following the tournament, a metatarsal injury suffered on club duty with Twente saw Bjelland miss a year of international football and he made his return in a 2–1 friendly win over Georgia on 5 June 2013. He scored his second international goal in a 6–0 2014 World Cup qualifying thrashing of Malta on 15 October 2013. Bjelland made regular appearances through 2014, but a knee ligament injury meant that failed to appear for Denmark between June 2015 and 11 November 2016 and he made his comeback playing the full 90 minutes of a 4–0 2018 World Cup qualifying win over Kazakhstan.
Bjelland was born to a Norwegian father and a Danish mother. His younger brother Thomas was also a footballer for BSV and Hellerup. He has two daughters.
FC NordsjællandSuperliga (1): 2011–12
DBU Pokalen (2): 2009–10, 2010–11
As of match played 19 September 2017.