C172 pilot reported they could not maintain control of speed or altitude when encountering strong downdrafts flying over mountains. They descended below Minimum Vectoring Altitudes and reversed course to avoid terrain.

Narrative

I departed ZZZ at approximately XA50L on an IFR flight to ZZZ1. Route of flight was ZZZ2 ZZZZ ZZZ3 and ETA was 1H 25M. I had been monitoring the weather for the past few days more concerned with icing conditions than strong downdrafts; but knew that at my altitude over the mountains; there would be a strong headwind and possible turbulence. Before departing; I thoroughly reviewed all available weather sources pertinent to the route of flight; including the AIRMET for turbulence over the area; on Foreflight. I began my climb out of ZZZ up to 10000 ft. as assigned by Departure. I encountered occasional moderate turbulence during the climb but after around 8000 ft. it cleared up. As I neared the ZZZ2 VOR; I was given a vector NE to climb above the MVA; then turn back toward the VOR.Groundspeed began dropping as I climbed through 9000 ft. and knew it would be difficult to reach 10000 ft. I requested 9000 ft. and the Controller amended my altitude to that. At 9000 ft. and approximately 25 miles from ZZZ4 on ZZZZ; I had entered IMC; my groundspeed began to reduce again; and my altitude began to drop as a result of strong; mountain-related downdrafts. The Controller requested I climb back up to 9000 ft. to comply with the MVA but my VSI was indicating a 1200 FPM descent; airspeed decreasing; and I could see on the Garmin GNS 430 TAWS yellow and red terrain pop up ahead of me. I immediately executed a 180 degree turn as my altitude fell below 7000 ft. and descended out of the clouds and back towards the valley and into lower terrain and VMC. The Controller asked if I was [requesting priority handling] to which I immediately replied yes; I am.Once clear of the terrain conflict; the Controller asked me to state my intentions to which I requested a turn south and towards lower terrain. Once it became clear that the higher terrain in the area would be a challenge for my aircraft; I was rerouted SW to the ZZZZZ intersection ZZZZ1 ZZZZZ1. Despite some lower groundspeeds; the aircraft was able to maintain an assigned block of 10000 to 11000 ft. Enroute to ZZZZZ; the Controller stated 'possible pilot deviation' and gave me a phone number to call; which I copied. Once I was able to join ZZZZ1; the mountain waves had subsided and the rest of the flight continued as normal. All systems in the aircraft including carburetor heat and pitot heat; were active and functioning normal. I never encountered a loss of airspeed indicator; altitude indicator; or a decrease in engine performance. I had decided to depart earlier in the morning as to prevent arriving during what was forecast to be the 'worst' part of the storm. For the route of flight; I knew I had ZZZ2; ZZZ4; ZZZ5 and several other airports that I could use to divert to if things had not gone as planned. I departed ZZZ with a full fuel and just myself as the sole occupant. I had flown into ZZZ Day evening VFR ZZZZZ1 and the flight was uneventful. I had studied the terrain and surroundings so I felt comfortable with the airspace and route of flight on the morning of the incident. What I had not expected; or had much previous experience with; were the strong downdrafts on the leeward side of the mountains. I had been educated on mountain flying in the past but my practical experience with it was rather limited. Now; experiencing it firsthand; I know how much of a factor it can be especially in a low-performance aircraft. I pressed on to my destination knowing that I was on an IFR flight plan with positive air traffic control; I had multiple divert points along the way; plenty of fuel along the way and that I could descend lower and that conditions would improve once west of the mountain peaks. From this; I have a newfound level of respect for mountain downdrafts; the importance of advising ATC; taking corrective actions quickly and decisively; and reinforced the helpful nature of ATC in aiding pilots in distress.