Since the Jernberg family moved to Kristiania it is hardly surprising it was from there Leonard emigrated April 19 1883. Final destination Chicago.
Digital Archives,
http://arkivverket.no/eng/content/view/full/629. Emigration link:
https://digitalarkivet.arkivverket.no/en-gb/gen/vis/8/pe00000000980884.
Perhaps he was going to join his sister Maria who had left for Chicago two years earlier.
Digital Archives,
https://digitalarkivet.arkivverket.no/en-gb/gen/vis/8/pe00000000952184.
Maria's death notice confirms that she was actually living in Chicago.
FamilySearch,
https://familysearch.org/ark:/61903/1:1:Q237-GC7J.
The Swedish (and Norwegian) patronymic surnames can be very confusing, but Ingela Martenius straighten out many question-marks in
Swedish names,
http://web.comhem.se/~u31263678/genealogy/Names.pdf.
Eventually the patronymic surnames became family names but the two possibilities both existed for a period.
Inger Olava could not have been called Trondsen in Norway since this means Trond's son as a patronymic (her patronymic should have been Trondsdatter, meaning Trond's daughter) and it could not be a family name since the only possibility for this is Jakobsen. Because Inger Olava's father was Tron(d) Jakobsen according to
FamilySearch,
https://familysearch.org/ark:/61903/1:1:NWND-WP2, and
Digital Archives,
https://media.digitalarkivet.no/en/kb20061108330647. (For some reason the fathers in this source are called -sson, which is a Swedish spelling.) In USA Olava seems to have used Jacobson as a family name.
According to the message above these are Leonard's parents at the time of the emigration to Norway (
Arvika västra landsförsamlings arkiv AI:20, 1872-1877, page 498):
Jan Jernberg, born May 16 1829, Älgå parish, Värmland
Maria Andersdotter, born December 26 1824, Eda parish, Värmland
Jan is born out of wedlock, the birth notice says his mother is Marta or Märta Jansdotter and that his father is believed to be Magnus Nordström. Source
Älgå kyrkoarkiv C:5, page 198.
Maria's parents were Anders Andersson and his wife Kerstin Olsdotter. Source
Eda kyrkoarkiv C:7, 1797-1845, page 274.