GA pilot reported while transiting MSP airspace they had a near miss with an air carrier on departure climb. Reportedly; the Air Traffic Controller was handling traffic from several airports and may have vectored the pilot incorrectly.

Narrative

I was on final approach into STP from refueling at ZZZ; coming in from ZZZ1. My destination was STP. My plane is a light but fast Experimental. The approach controller told me he was going to take me over MSP to get to STP from the west. The evening (it was about sunset) was clear. I was under vectors at 4000 feet. The course I was assigned was taking me a bit south of MSP; by perhaps three or four miles. Visibility was good. I was watching for traffic on ADS-B and saw a lot going into and out of MSP. I saw an air carrier (XXX callsign but I did not get the number) taking off from MSP southbound. I saw that we might be in conflict; but thought that he should be limited to 3000 feet; as I was at 4000 feet and headed across his track. The moment that I saw his climb went past 3000 feet; I turned north to avoid a potential conflict. Soon after I started my turn; my controller gave me an emergency turn 90 degrees to the left (360 degrees) as he had realized his mistake; apparently. My suggestion for the controllers around MSP is that if you are crossing an IFR across MSP to STP; you should either cross them right over MSP; or have the MSP departure limited initially to 3000 feet.I think that that my controller; who was handling non-MSP traffic; was rather overwhelmed by a lot of VFR traffic into the area. He was giving IFR and VFR instructions to a number of little planes and missed his mistake to send me into conflict with the air carrier. I believe our closest approach was 700 feet and perhaps two miles; but I was turning to avoid a collision and did not closely check the miss distance. I had the air carrier visually.