Bezjak made his debut for Celje on 9 April 2008 against Maribor, coming on as a substitute for the last six minutes. The following season he made his first start for Celje, on 2 August 2008, in a 0–0 home draw against Domžale before being substituted by Saša Bakarić. Two weeks later, Bezjak scored his first goal in a 2–2 away draw against Primorje. He began to establish himself in the Celje first team from the 2009–10 season, making 24 league appearances and scoring 6 goals.
On 22 August 2012, Bezjak signed a four-year contract with Bulgarian A Group side Ludogorets Razgrad. He made his debut for Ludogorets against CSKA Sofia on 22 September where he came on as a substitute in a 1–0 home win. His first goals for Ludogorets came on 3 May 2013 when he scored twice in a 3–0 home win over Chernomorets Burgas. He made 14 appearances during the 2012–13 season, finishing with 5 goals. In the following season Bezjak became a first team regular. He scored his first goal of the season in the second league game, a 1–0 win at home to Chernomorets Burgas on 27 July 2013. On 27 October 2013, he scored his first-ever career hat-trick, netting four goals in a 5–1 league win over Lyubimets 2007. He also scored three goals in two matches against PSV Eindhoven in the group stage of the Europa League. In February 2014, he netted in both legs of the 4–3 aggregate victory over Lazioin the round of 32 of the Europa League. On 13 March2014, he earned а penalty kick in the 3–0 home loss against Valencia, but missed from the spot. On 15 May 2014, Bezjak scored the winning goal in the 1–0 victory over Botev Plovdiv in the 2014 Bulgarian Cup Final to help his team achieve a double. He ended the season as the club's top scorer, scoring 20 goals in all competitions.
Rijeka
On 14 May 2015, HNK Rijeka announced that Bezjak had signed a three-year contract, tying him with the club until June 2018. He made his official début for the club on 10 July 2015, in a goalless away draw against Inter Zaprešić in Round 1 of the Croatian First Football League. On 19 July 2015, Bezjak scored a brace in a 3–3 home draw against Slaven Belupo. With 13 goals to his account, he was the club's top scorer during the 2015–16 season.
On 1 February 2018, Bezjak signed a two and a half year contract with Jagiellonia Białystok. His debut for Jagiellonia was against Piast Gliwice in a 2–0 victory being subbed in the 73rd minute. Bezjak's first goal for the club was on 23 February 2018 against Lechia Gdańsk in a 4–2 win at home.
International career
Bezjak earned his first cap for his country on 14 August 2013, after coming on as a substitute in the 2–0 loss against Finland in a friendly match. On 23 March 2016, Bezjak scored his first goal for Slovenia in the 1–0 win against Macedonia. On 8 October 2017, Bezjak scored twice for the 2–2 draw between Slovenia and Scotland.
