It's a polarity problem, and it's baked into the layout itself. What you've built is a reversing loop. When both turnouts are lined up, the inner loop becomes a path that wraps back around and meets the outer loop again, and that path is electrically backwards compared to the rest of the track. Left rail meets right rail somewhere, the power pack sees a dead short, and it cuts out. One turnout alone doesn't complete the loop, which is why it runs fine untill you throw the second one.
The fix is to isolate the section of track between the two turnouts from the rest of the layout, then feed that section with its own wire and a double pole double throw toggle switch. The toggle lets you flip the polarity of that section independently. With the train inside the isolated section, you throw the toggle so the rails match whichever track it's about to run onto, and off it goes the other way.
The annoying part with EZ Track is the roadbed is molded right into the track sections, so you can't just pull a plastic joiner out of a gap. You'll need to cut gaps in both rails on a piece of track at each end of the reversing section. A razor saw does it, just cut clean through the nickel silver on both rails and you're done. Two cuts at each end, four total. Then solder feeders to the isolated section and run them through your DPDT switch back to the power pack.
One thing to keep in mind with a DC pack: flipping polarity reverses the loco's direction. So the routine is run the train into the isolated section, bring it to a stop, flip the toggle, then open it back up and it heads out the far end going the opposite way. That actually matches what you said you wanted in the first place, a way to turn the train around. If cutting up track sections sounds like more than you want to do right now, the alternative is to wire the two turnouts so you never throw both at once, but that kills the whole reversing idea you were after.