Shah, 35, former mayor of the capital city of Kathmandu, has dominated the race to become Nepal's PM during the campaign, gaining popularity across the country.
Shah's centrist Rastriya Swatantra Party, which was founded three years ago, was leading in 37 seats while former Prime Minister K P Sharma Oli's Communist Party of Nepal (Unified Marxist-Leninist - UML) was leading in three, early counting trends from the election commission showed on Friday morning.
The centrist Nepali Congress led by 49-year-old Gagan Thapa, the country's oldest party, was leading in five seats.
Results were likely to be clear by Friday evening or Saturday, officials said.
Out of a total 275 seats in Nepal's lower house of parliament, counting was in progress for 165 seats that are elected by direct election, while the remaining 110 will follow a system of proportional representation.
Shah drew large crowds during his election campaign, connecting with legions of young voters clamouring for change on the ground and online, even as he took on Oli, 74, on his home turf in the Jhapa 5 constituency along the Indian border.
Perched between China and India, the country of 30 million people has been plagued for decades by political instability, rippling a largely agrarian economy and worsening unemployment - structural issues compounded by rampant corruption.
The long-festering malaise erupted into street demonstrations last September, triggered by a social media ban, that brought thousands on the streets, leading to clashes and fatalities that forced the resignation of Oli.
