Solem played end at the University of Minnesota for Henry L. Williams from 1910 to 1912. He then began his coaching career, coaching for a pre-NFL professional football franchise called the Minneapolis Marines. He coached there for three years from 1913 to 1915 before coaching a few years of high school football at East Des Moines High School and South High School of Minneapolis.
In 1932, Solem signed a three year contract to succeed Burt Ingwersen as the 13th head football coach at the University of Iowa. He took over an Iowa football program that had recently been suspended from athletic competition in the Big Ten Conference for a month. More importantly, the Hawkeye program was suffering from the effects of the Great Depression. Since Iowa was a predominantly agricultural state, the Depression hit the Hawkeye athletic program particularly hard. The school could not even afford to pay Solem his full salary when he was first hired.
After winning the opening game of the 1932 season, Iowa lost their final seven games of the year. In the off-season, Solem loudly complained that athletes were being discriminated against in Iowa City and were not being hired for normal student jobs, probably as a result of Iowa's recent conference suspension. An independent investigator substantiated Solem's claims.
Hopes were not high in 1933, but Iowa responded with a 5–3 final record and Hawkeyes quarterbackJoe Laws was named the Big Ten MVP. Solem's contract was extended two years, and with the departure of Iowa's athletic director, Solem was appointed to that position as well. Meanwhile, he struggled with a lack of cooperation from fellow Big Ten schools in schedule meetings. Each conference member was supposed to be guaranteed five conference games, but due to Iowa's recent suspension, other Big Ten schools were more accommodating to Notre Dame, a non-conference school, than they were to the Hawkeyes. Solem fought tirelessly to have Iowa regarded as a member of equal standing within the conference.
After two season opening wins, Iowa failed to win any of its final six games in 1934, and a season that started optimistically ended with a 2–5–1 record. Iowa managed to bounce back the following year with a respectable 4–2–2 record in 1935 behind the play of captain Dick Crayne and the sensational Ozzie Simmons. After the 1935 season, Solem worked with the president of the university at the time to improve the job situation for athletes in Iowa City. Their reforms helped Solem bring in what Solem called "the finest group of freshmen during my tenure". Many of the athletes Solem successfully recruited to Iowa would become the foundation for the 1939 Hawkeye team, nicknamed the "Ironmen", which included Nile Kinnick and Erwin Prasse.
A conflict between Solem and Ozzie Simmons overshadowed the 1936 season, which ended in a 3–4–1 record. After a 52–0 loss to Minnesota, Simmons quit the team for a couple days. Simmons stated that he felt Solem had been too critical of him for Iowa's failures during the 1936 season. Simmons was convinced to return to the team a few days later.
Before the final game of the season, reports were heard that Solem was leaving at the end of the year. Iowa defeated a heavily favored, nationally ranked Temple team, coached by Pop Warner, 25–0, to end Solem's coaching career at Iowa. After the 1936 season, Solem announced he was leaving Iowa for Syracuse University.
Later, Solem wrote, "I went down to Iowa City for the Iowa-Minnesota game, the first time I had been on the campus since the year after we left Iowa. As I sat there watching the game and admiring the beauty and growth of the campus, recalling the many friends we had in that lovely town, I could not bring myself to have any feeling of dislike or hate, but rather a feeling of regret...and gratefulness for having once been a part of that great institution."
Solem coached at Syracuse from 1937 to 1945. During his time at Syracuse, he tutored a young assistant coach named Bud Wilkinson, who went on to coach the Oklahoma Sooners and win three national championships. After his time with the Orange, he coached at Springfield College in Massachusetts from 1946 to 1957, compiling a 58–33–7 record. Solem coached a total of 37 seasons of college football, with a combined record of 162–117–20 at five schools.
One of his former players said about Solem, "He was a better offensive coach than he was a defensive coach...He was a fine, clean living man, and he was very well thought of in athletic circles." Former player Ozzie Simmons remarked, "He probably could have been tougher (on the players), but he was the finest gentleman I've ever been around."