Flight Instructor reported an NMAC with a twin-engine turboprop aircraft departing PAAQ airport.

Narrative

While landing Aircraft Y announced they would be departing after the aircraft landing (Me) we do a touch and go; and begin climbing out at about 500 fpm when beginning to turn xwind the applicant in the right seat says oh! And begins taking evasive action to the right. I see the Aircraft Y and bank and climb to avoid the Aircraft Y; climbing towards us and appearing to not take any corrective action. We were on a collision course for what felt like a few seconds until our turn started and we narrowly avoid the other aircraft. Following the Aircraft Y's passing I released back pressure to stop the climb and turned forward to return to level flight following the resulting pitch low and right bank. The rest of the flight was uneventful. There were no further communications heard from Aircraft Y and we made position reports at the typical points. I feel that they did not leave adequate spacing give the types of aircraft involved but I understand the possibility that I could have missed the call they were taking off and evasive action could have been taken sooner. I was still waiting to turn x wind for an aircraft ahead of myself as that spacing was inadequate for the planned short approach.