C182 pilot reported they descended due to a NMAC encounter and then received terrain and obstruction alerts before climbing back to an appropriate altitude.

Narrative

VFR takeoff North from ZZZ; we requested a published transition before a SW departure. Tower approved. We proceeded Northerly along the East side of the river; about 1500 msl. After passing the visual waypoint; we executed a procedure turn to make a Southbound pass before departing the area.Nearly completing the left turn; we spotted a small high-wing ASEL Southbound above; less than 500ft vertical and maybe 400ft horizontal. We were unable to reverse our turn without conflicting with downtown buildings; so we increased our turn and descended to improve vertical separation. However; we were now below our intended minimum altitude; bottoming out at 900 msl; 500 agl. The conflicting traffic was now above and behind us; flying the same direction; which is completely blind to us and likely blind to them. We received numerous terrain and obstruction alerts from onboard systems. After flying away from the immediately concerning obstructions; we contacted ZZZ Tower informing them of the concerning traffic and asking if we are able to climb away. Tower confirmed traffic was now 1/2 mile behind; so we climbed and continued on our departure.In reviewing the events later; on the ground; I am surprised that Tower never made any proactive traffic advisories to either aircraft within their airspace. Even stranger; when we departed the Delta airspace and advised Tower; we received no reply nor instruction to change frequency.Regarding my own decisions; I failed to see and avoid the conflicting traffic earlier. It's a busy and cluttered area; so visually recognizing traffic of course is difficult. However; I failed to use my onboard ADS-B In traffic monitor to its fullest potential. I should have been more actively monitoring that traffic page and including it in my scan; particularly since my pre-flight briefing noted this is a higher traffic area than I usually fly in. Further; when I spotted the potential conflicting traffic; I probably could have aggressively-reversed my turn. This would have had the distinct benefit of avoiding positioning another aircraft above and behind me. Given this experience; I would be more inclined towards that response in a future similar situation.