Light aircraft pilot departing BJC airport reported an NMAC with another light aircraft.

Narrative

Upon departure from Runway 30R at BJC; I; along with a CFI; were cleared to turn northbound 2-3 miles off the departure end of the runway. At about 1 mile off the departure end of the runway; Tower cleared us for a northbound (right) turn. When we turned northbound; a [light aircraft] in our blind spot was less than 200 feet off the nose of our aircraft. I steepened the right hand turn; and the [light aircraft] also took evasive action turning hard right. After listening to ATC audio on the ground; the [light aircraft] was cleared for a midfield right downwind to 30R. Instead; he was flying through the departure leg at about 6;300 feet MSL. This was unnoticed by ATC until after the near collision occurred. The [light aircraft] was being flown by a student- solo pilot.After the near miss; the [light aircraft] pilot said he became 'disoriented' looking for traffic. ATC asked if he had the airport in sight; because after the incident the [light aircraft] was now flying away from the airport on the departure leg. I do not think the Tower Controller ever realized the mistake; nor was the [light aircraft] pilot instructed to call the Tower.I believe Tower may have been 'behind the plane'. It seems they assumed the inbound [light aircraft] was adhering to their clearance without verifying it. I believe the student solo pilot either thought he was landing on 12L (the reciprocal runway); or believed he was cleared for a downwind for 30L and was attempting to fly a crosswind for this runway. The [light aircraft] pilot did eventually land on 30L.