2010-11: Conner Bleackley had nine points in five midget AAA games with the Keystone Raiders in Alberta — and was one of the top scorers for the Okotoks Oilers bantam AAA team. He scored 7 goals with 2 assists and had 4 penalty minutes with the Raiders. In 29 bantam games he scored 36 goals with 32 assists and had 62 penalty minutes. Playing with the South Senators in the 2011 Alberta Cup he scored 3 goals with 2 assists. Bleackley was selected by Red Deer in the first round (21st overall) of the 2014 WHL Bantam Draft.
2011-12: Bleackley skated in 16 WHL games with Red Deer as a 15-year-old — spending most of the season with the UFA Bisons midget AAA team in the Alberta league — and won a gold medal with Team Alberta at the 2011 Western Canada U16 Challenge Cup. Bleackley scored 2 goals and was -5 with 6 penalty minutes skating for the Rebels. In 26 AMHL games with the Bisons he scored 13 goals with 17 assists and had 41 penalty minutes. Bleackley scored 2 goals with 2 assists in four games for Team Alberta.
2012-13: Bleackley played in the WHL with Red Deer and represented Canada Pacific in the 2013 U17 World Hockey Challenge. He scored 9 goals with 9 assists and was -4 with 28 penalty minutes in 66 regular season games for the Rebels. Red Deer finished third in the Central Division and reached the second round in the playoffs. In nine playoff games Bleackley scored 2 goals with 1 assist and was -1. An assistant captain for Canada Pacific, he scored 3 goals with 1 assist and had 12 penalty minutes in five games at the WHC.
2013-14: Bleackley was named team captain for Red Deer in his second year with the Rebels.