I want to have a mountain range in my continent that splits it into two parts, such that it is incredibly difficult for them to communicate. How could I pull this off in a plausible way without having to resort to magic?
Doesn't this already happen in real life?
The Urals separate Asian Russia from European Russia, the Himalaya separates the Indian subcontinent from Asia, just to cite some examples.
When two tectonic plates with continental crust collide, they will end up forming exactly what you ask: a unique continent separated by a mountain range right at the point of crash.
Double Mountain Range, with Desert
As in the real world, simply have two large tectonic plates forcefully collide. Make the mountains as tall as the Himalayas, and as long as the Andes. Make the mountain range extend all the way to mostly uninhabitable Arctic/Antarctic regions.
A secondary up-thrust from a secondary break in the tectonic plates would make a second mountain range mostly parallel with the first. Between is a barren, dry basin like the Atacama desert, mostly devoid of habitable life.
Only crazed, desperate explorers would be willing to travel for weeks through these extreme environments. In reality, though, people are extremely tenacious. Some adventurous nut job is going to try to cross the mountains and the desert, and eventually will succeed.
Perhaps throw in some very nasty wild monsters/aggressive mountain predators. Or, even better, some vicious parasitic creature that dwells in the deserts. The first night you sleep in the desert, it latches onto your shoes, crawls up to your leg, and then burrows into your nervous system. It injects chemicals that slowly make you lose your mind. Everyone that comes out of the desert is a raving lunatic.
Add local legends stating that evil beings reside in the mountains, and you have social reasons on top of biological reasons to shun the journey.
The Ocean Work-Around
Unless the mountain range runs across the earth, one side will still end in Ocean. By adjusting the ocean and wind currents, you could have a massive desert on one end of the continent. So desert on one side, frozen wasteland on the other. Prevailing winds circle away from either direction of a long peninsula of desert/mountain jutting out into the ocean. Without advance ship technology, no one will get past these straits of death.
To be clear, it took renaissance technology for Europeans to circle around just the northern coast of Africa. So, this is a reasonable possibility.
What you're looking at is one of two things the more likely is what we call a suture zone, these are formed when oceans close due to the subduction of the spreading centre that creates the ocean floor and thus the eventual subduction of the ocean basin and collision of the continents on either side. In the modern world the existing sutures are quite old mountain ranges; the Urals in Siberia and the Appalachians in the US being the best examples. Sutures run the full length of the old continental margins so they're highly likely to bisect the resulting combined continent entirely and quite neatly.
Less likely but still possible is a mountain range built by a continental convergence zone here we have several active examples, the Atlas range in north Africa, the Pyrenees and Alps in Europe are all the result of the collision of Africa with Europe, and the Himalaya in Asia caused by India's collision with the Asian continental plate. These mountain ranges are far more dependent on the local geography and geology in terms of extent and morphology and so are far less likely to bisect a continent entirely or neatly.