Student pilot with Flight Instructor reported airborne conflicts and unsafe situation with several other aircraft in traffic pattern. Flight crew exited the pattern and diverted to nearby airport.

Narrative

I was performing an IFR cross-country training flight with an authorized instructor aboard. We were on an IFR flight plan to GZH. After switching to GZH CTAF; we canceled IFR and proceeded with the approach into Runway 19 under VFR. However; shifting winds favored Runway 28; so we planned to circle. An 8-mile radio call was made to advise any traffic of our location and intentions. At this time someone advised us of traffic in the pattern for Runway 28. We decided the safest pattern entry would be to cross overhead and teardrop into the downwind for Runway 28. We proceeded inbound and made 5-mile; 3-mile; and overhead radio calls. We flew 4 miles out and then made a radio call while tear-dropping into the downwind and descending to pattern altitude. At this point; we heard the first radio call from another aircraft in the pattern; stating they were turning left crosswind Runway 28. I had that traffic insight and noticed another aircraft just off of the runway. We turned into the downwind and made another radio call. In the downwind; we recognized these aircraft were faster than us. One aircraft said they would remain higher to deconflict. We communicated that we would extend our downwind; allowing the faster aircraft to turn base first; and included our altitude to increase the situational awareness of all aircraft. We felt the situation was unsafe and weren't comfortable continuing our approach while pilots of different speed aircraft weren't communicating their positions in the pattern. We exited the pattern and diverted to a nearby field. We made a radio call that we were exiting the pattern and stated our direction of flight. This situation was caused by a lack of radio communication from pilots in the pattern. To prevent a recurrence; frequent radio calls should be made from all aircraft to enhance sequencing; improve situational awareness; and increase safety at untowered airports.