"steep mountainside" is correct and usable in written English.
You can use it when describing a mountainous landscape. For example: "The sun shone over the steep mountainside, and a flock of birds flew by."
To my left was steep mountainside; to my right, river rapids.
The railway was built to take slate down the steep mountainside to the sea to ship across the world.
A little farther, thick columns of smoke poured from vegetation on a steep mountainside and spiraled into the sky.
While I started up the steep mountainside the children bounced ahead of me, teasing me that I moved slowly for someone so tall.
His white villa, full of fine old paintings, statues and mosaics, clings to the steep mountainside overlooking the Gulf of Salerno.
But what is remarkable is what lies across the road – a steep mountainside, terraced with layer upon layer of deserted stone buildings.
Blue-painted buildings line the streets, clinging to the steep mountainsides.
The Herdwicks, Mr. Brown said, are a special breed uniquely able to forage on the steep mountainsides.
Eventually, we drove up onto bleak, washed-out roads where, through the scratched plastic windows, we anxiously watched the steep mountainsides for rock slides.
In summer, its steep mountainsides shimmer with wildflowers, and glacial rivers irrigate small valley farm fields and orchards, which yield generous crops of peas, potatoes, apples and plums.
Days of rain unleashed tons of earth, rock and torrents of water on Wednesday down steep mountainsides, directly into towns that are weekend getaways for the Rio area.
