General aviation pilot reported multiple traffic conflicts in the traffic pattern at a non-towered airport on a training flight with an instructor. The pilot maneuvered away and slowed; to provide separation from another aircraft not following non-towered procedures.

Narrative

Flying to obtain Private Pilots License with CFI; entered pattern at 2W5. Upon touch and go from second pattern making calls on 122.7 CTAF; reported on the go from Runway 02 for left closed traffic; announced left crosswind for Runway 02 upon crossing 700 feet MSL.Flight instructor of aircraft 1 called out that another aircraft took off behind us unannounced and cut a crosswind at 400 feet MSL. As we leveled out in the crosswind; pilot on the controls on left side identified aircraft number 2 that was approximately less than 500 feet horizontal and about 300 feet below us. (at XA:47; time of incident via on ADS-B exchange).Called them on CTAF asking why they cut us off. Pilot on comms responded with 'we were told to turn crosswind at 400 feet.'Crew of aircraft 1 extended crosswind and slowed back to increase space to turn number 2 in downwind. Made calls over CTAF to explain intentions to prevent further complications. Second pilot of aircraft 2 finally called over the radio to confirm they had heard our intentions; and apologized for turning in front of pattern already in progress/not making correct radio calls or traffic procedures.Contributing factors involved multiple aircraft in traffic pattern and taking runway while other aircraft were landing or had not exited the runway to allow for aircraft to depart; all with erratic traffic patterns. Their instructors were seemingly oblivious to multiple FAR violations on runway; poor traffic pattern practices; and inaction on radios for communication.