Pilot reported two separate conflicts with aircraft while conducting pattern work in a crowded traffic pattern at the CDC airport. Busy general aviation traffic; flight training; helicopter operations; and regional airline traffic contributed to frequency congestion and created a difficult environment for safe operations.

Narrative

1st incident. While conducting closed traffic pattern work to practice power off 180 landings; short field takeoff and landings; wheel landings; 3-point landings; we encountered two separate incursions by local flight school training aircraft in the vicinity. Cedar City airport has become overly saturated with training of fixed wing and rotorcraft all sharing the same frequency and runways/taxiways which has become dangerous due to the sheer volume of aircraft as well as multiple different ops within the airport boundaries. The closed pattern work is challenged by practice ILS; VOR and RNAV approaches; which are not usually an issue except when the incoming long final aircraft don't update their position reports as they arrive from 8 to 12 miles out. There are regional jets; charters; private jets all operating without issues as they all use accurate position updates. Frequent frequency congestion occurs due to the additional large qty of helo ops at the field. To say this is a challenging and confusing environment would be putting it gently.Our incident occurred during our initial departure from runway 20; inbound ILS practice traffic; Aircraft Y called final approx. 5 miles out; we spotted them visually while we held short for their landing. Upon their crossing the threshold at approx. 200 ft; they called missed and sidestepped 500 ft. west of runway as is expected. We then rolled out to departure position; then as they approached midway down the 8400 feet long runway; we started rolling our departure; all while using radio calls at each location and movement of our aircraft. To both our surprise; our aircraft accelerated and climbed far faster than either the Aircraft Y pilot or I expected. In no time we were abeam Aircraft Y at same altitude and approx. 500 ft. laterally. We continued our climb quickly to 7000 ft; which is above pattern altitude of 6400 ft. and their altitude of approximately 6000 ft. During that time Aircraft Y commented that we were too close for them. We acknowledged full visual the entire time and apologized for the closeness while continuing to climb well above their flight trajectory and altitude. Approximately 1 mile past departure end of runway we turned right crosswind; then downwind well above and away from the departing Aircraft Y as they exited the area southwest bound. No further interaction or communications ensued. End of incident. Conclusion. Underestimated performance of Aircraft X on departure; underestimated slowness of departing/climbing Aircraft Y. Aircraft Y was following procedure well; we simply misjudged our performance differences. 2nd incident 15-20 minutes later. Pattern has now continually been filling up with more training aircraft and helicopters from the flight school. Generally once more than 6 to 8 aircraft are in the pattern it becomes difficult to mentally keep track of where everyone is; especially considering the helicopters operate opposite pattern and opposite side of airport but call out the same runway; taxiway and intersections fixed wingers use. It can be a challenge to understand exactly where the helicopters are. Suggest a different frequency and keep them well away from the fixed wing traffic due to their nonstandard maneuvers and their incompatibility with the fixed wings. During a short approach while mid field down wind; we called position and plans for short field approach additionally asking if anyone on long final. No response; no visual conflict; all appeared clear. At approximately abeam numbers we executed right base at 6400 ft. pattern elevation and called position report on radio; again no response or other calls announcing inbound practice ILS from the same Aircraft Y. As we rolled out on base; perpendicular to final approach course I spotted Aircraft Y approx 500 ft. below and 1500 ft. ahead of us. We were headed directly towards them at this point as they were on a short final. Again; no radio calls heard from Aircraft Y. We turned final behind them; fully visible at all times; and announced our position and intent for landing and we were on short final for 20. We realized we were far too close behind them to allow adequate spacing and were closing quickly so we broke off descent; sidestepped 500 ft.to west and called missed approach. Upon climb out while overtaking Aircraft Y and far west and above their position they again commented it was too close for them while they continued to land. We apologized and said we'd open things up a bit more for spacing. I should have asked them why they didn't make several position reports while on such a long final as is customary when long finals mix with closed traffic. It's possible they did; and we missed the calls considering the chaotic radio transmissions at this airport. Never were either of the two incidents anywhere near close enough for a collision; it certainly made the training aircraft uncomfortable. We on the other hand are much more used to flying close patterns; short field ops; and very maneuverable aircraft so as this did not seem like anything dangerous or alarming; simply a need to adjust flight track to work with everyone nearby. The Aircraft X conventional gear airplanes are very agile; high-performance machines with outstanding visibility and handling characteristics. I've also owned and flown similar Aircraft Y types enough to understand their visibility and performance shortcomings. Conclusion. Too many aircraft training on too many different types of operations in this area. Better radio work by all parties could help; possibly a separate frequency and location for the helicopter operations. Certainly; better and more frequent position reports to include extra-long finals. 5 to 12 miles out are frequently heard over radio. Cedar is a wonderful; long and open airport with great NAVAIDS; a crosswind runway; taxiways and favorable winds. The traffic pattern has been challenged by the large flight school oversaturating the area with training aircraft of mixed types. We are all aware of our duty to share the airspace and dutifully do exactly this. I would suggest a closer look of the types of mixed ops taking place here and possibly introduce new protocols to help manage the traffic. Thank you for considering this report in our ever-continuing search for safer airspace for all.