General aviation pilot reported a NMAC at a non-towered airport in the traffic pattern. Pilot took evasive action to avoid a collision.

Narrative

In Aircraft X; with a DPE on a commercial ASEL practical test flight; I departed Runway XX at ZZZ. There were several other aircraft on the CTAF either in the pattern; preparing for departure on the ground; or en route to land a few miles out. At the time I departed; I recall hearing Aircraft Y on the CTAF state they were several miles to the south and planning to join the left downwind for runway XX. I saw that aircraft on the ADSB (ForeFlight) and confirmed their location. After departure; I turned left crosswind at 700'; then shortly thereafter left downwind for a westerly departure. I made CTAF radio calls for both turns. Very shortly after I called left downwind; I recall hearing Aircraft Y call left downwind as well. I knew they were very close behind me and must be moving faster than me. We were showing the same altitude on the ADSB. When I was about midfield or abeam the numbers still on the downwind I made another radio call just as a reminder that I was on the downwind and would be departing to the west. I made a slight turn to a 270 westerly heading at that time since final was clear and the other other traffic in my immediate vicinity was right behind me. I realized they were moving faster so figured that slight left turn would help deconflict the traffic. Moments later Aircraft Y called turning base. I saw on ADSB they were very close and turning into us. I asked 'Aircraft Y do you see us? Aircraft X.' They responded with 'negative.' I made an immediate turn to the south and started a climb and told them such. By this point I was well southwest of the final approach course/extended centerline. It appears they flew past that or turned extra wide on their base to final turn and became extremely close to me. I believe we were at the same altitude during the entirety of our shared downwind leg. We were both making CTAF calls; so I do not understand why they were so close or if they truly never saw my airplane or if they were not listening to the CTAF and did not hear me; or see me on their ADSB. I never was able to make visual contact with that plane. I believe this might be an issue of me being in a high wing and them being in a low wing plane. Perhaps if they were slightly above me they could not see. Regardless of the possible reasons; it was quite unsettling as they turned into me and got very close to a mid-air. It was extra stressful as I had a DPE in the right seat as this was a commercial ASEL practical test for me. Happy to answer any questions you may have. Thank you!