Xjyuk

I've pretty much finished the map of a world I'm building for a fantasy novel, but I've realised that on one of the continents I need there to be forest on both sides of a large mountain range. I've never made maps/worlds before so have had to research everything from scratch and I didn't really account for rain shadows.

The climate in this area is humid continental. The people living on the right side of the mountain have to some extent used magic to help the forest grow, but I would like the forest to exist naturally rather than have it all be down to magic.

The world is like Earth, only the land masses are different.

Is it realistic for there to be a forest to the right of this mountain range, or would this be in the rain shadow and too dry for a forest? If so, are there any other conditions I could alter to make it work?

I'll include a picture (with my crude climate colouring overlaid, though still unfinished on the other continent):
(Some of the lakes are black, because I realised I had too few lakes and rivers and went back to add more.) I've circled the area I'm asking about. I had planned for both sides of this mountain range to have forests, but more so on the east side.

Yes, depending on wind patterns.

It's definitely possible for a mountain range to have significant precipitation, so long as there isn't a strong, dominant wind pattern going across the mountains.

Globally, climate patterns look like this:

Looking at the US, in much of the West there are large rainshadow regions. These are behind the Cascade and Rocky mountains, because there's a strong, persistent wind system from East to West. Further North, though, areas like Montana are heavily forested, despite being "behind" the mountains, because they get moisture from air currents from the North, as well.

Wind patterns on Earth are shown here. They change seasonally, in many areas, so fixed maps aren't as useful. (That map shows wind patters as they are in real time, I think.)

The Appalachians don't have any significant dry areas surrounding them, in part, because there are no major cross-mountain wind systems. Major storm systems come from multiple directions, so the effect of the mountains in regulating the climate is severely reduced. (Plus, they aren't terribly tall, so their rain shadow effect would be small, even in the case of strong prevailing winds.)

Many mountain ranges in the far North, such as the Urals in Russia, don't produce deserts because Arctic air currents provide both sides of the mountains with different moisture-rich wind systems. For the Urals, winds from the West bring warmer air from across Europe, while winds from the East bring cooler (but still relatively moist) air from across Siberia.

On your map, you could easily have wind systems dominated by Arctic winds on at least one side of your mountains, perhaps with a dominant wind system coming from the South in the other direction. Your climate pattern would be similar to that present in the Ural mountains, with forest present on both sides, but with one side markedly colder than the other.

Yes. Example: In western North America, there are forests of arid tree species (Juniper, Lodgepole Pine) in the rain shadow of many mountain ranges. The trees are slower-growing than their wet-side counterparts, the forests thinner and more sparse-looking.

You may not need to look any further than the Andes and the Amazon basin to find an example on Earth. There is dense rain-forest "to the right" of much of the Andes mountains.

If you want forest on both sides then there is the Urals with boreal forest on both sides (Urals are obviously lower than the Andes, but still a significant mountain range).

If you look at the western end of the European Alps, they clearly hook around to the South in the vicinity of northern Italy. The natural vegetation on both sides of the North-South section is deciduous forest.