General aviation Pilot reported a near miss with another aircraft while in a congested traffic pattern at a towered airport.

Narrative

Approaching CFO (Space Port); contacted Tower from 10NM E-SE. Tower instructed me to enter and report 5NM straight in final for 26. Approximately 1 or 2 Miles south of intercepting final at 5NM out on a NW heading; Tower instructed me to turn east (outbound) for sequencing. Almost immediately following that direction; they gave me a traffic alert. I reported 'Negative Contact' as I did not have traffic in sight. Seconds later; I acquired traffic immediately below me by ~100ft and less than .25 NM. Impact was narrowly missed. In review of ADS-B data on FlightRadar24; the traffic I nearly collided with did not show up anywhere near my vicinity. However my onboard avionics did pickup the Transponder Mode C signature for the traffic during the turn. I don't recall seeing that particular aircraft on my avionics until in mid-turn however lots of other traffic were depicted on in the pattern and around CFO. During the turn; my avionics began providing collision warnings.Tower was talking to the conflicting traffic throughout the entire duration. To me; Tower was saturated with only 1 person handling both Ground and Air frequencies. There were lots of planes in the pattern with little room between Class D and B airspaces.I believe the instruction to turn me >90 degrees right in such close proximity to traffic approaching quickly from behind and to the right left that plane obstructed from view until nearly to late to avoid. Compounding this was that the traffic alerts were not given with immediate concern nor were they given upon initial instruction to turn the aircraft towards a collision trajectory.