Lindsay, like his father Troy, played running back in high school. At South High School, Lindsay elected to wear the same jersey number as his father, 22. Playing for the Rebels, he chased the Denver Public School District record for rushing yards (4,400), which was set by his father in 1979. After a successful first three years of high school, Lindsay committed to playing for the University of Colorado Boulder. Shortly after surpassing his father's rushing record during his senior season, Lindsay suffered a severe ACL tear that brought an end to his high school career. Despite this injury, Mike MacIntyre, Colorado's coach, upheld the school's scholarship offer for the 2013 season.
Before the 2014 season began, Coach MacIntyre began calling Lindsay the "Tasmanian Devil" due to his quickness and determination on the field. As a backup running back and kick returner back during the season, Lindsay played in all 12 of the school's games. He led the team in kick returns and return yards and finished fourth on the team overall in rushing yards. Lindsay averaged 4.9 yards per carry in his 79 carries of the season.
Like the 2014 season, Lindsay saw action in all of the Buffaloes' games. He led the team in rushing attempts (140) and rushing yards (643), averaging 4.7 yards per carry. Lindsay scored seven touchdowns during the season, 6 rushing and 1 receiving. Despite only being a sophomore, Lindsay was elected by his teammates and coaches to be a team captain.
During his senior season in 2017, he rushed for a career-high 281 yards against Arizona. During the 2017 regular season, he ranked ninth among all Division I FBS players with 1,474 rushing yards. Lindsay finished the 2017 season with the most rushing attempts (301) in Division I play. He was selected as a semifinalist for the Doak Walker Award for the nation's top running back.
Lindsay set the Colorado career record for all-purpose yards and yards from scrimmage and ranked second in school history in rushing yardage and fifth in points scored.
Lindsay signed with the Denver Broncos as an undrafted free agent on May 1, 2018. He made the Broncos' final roster after an impressive preseason, and was named the third running back on the depth chart behind Royce Freeman and Devontae Booker. In Week 1, against the Seattle Seahawks, he rushed for 71 yards on 15 carries and recorded two receptions for 31 yards and a receiving touchdown. The following week, in a 20–19 victory over the Oakland Raiders, Lindsay ran for 107 yards and had one reception for four yards. He became the first undrafted player in NFL history with 100+ scrimmage yards in each of their first two games. During Week 3 against the Baltimore Ravens, Lindsay was ejected for throwing punches after a fumble. He was held to 20 rushing yards as the Broncos lost 14–27. In Week 4, on Monday Night Football against the Kansas City Chiefs, Lindsay had 69 rushing yards and added another rushing touchdown. In Week 11, in a 23–22 victory over the Los Angeles Chargers, Lindsay recorded his first multi-rushing touchdown game. In a Week 12 victory over the Pittsburgh Steelers, he had a season-high 110 rushing yards and a rushing touchdown. In a Week 13 game against the Cincinnati Bengals, he rushed for a career-high 157 yards and two touchdowns, earning him the AFC Offensive Player of the Week award.
On December 18, 2018 Lindsay was voted to his first Pro Bowl making him the first undrafted offensive rookie in NFL history to be voted to a Pro Bowl. On December 24, 2018, Lindsay, who injured his wrist on Monday Night Football against the Oakland Raiders, was placed on injured reserve ahead of the Broncos’ final season game. He finished his first season with 1,278 total yards (1,037 rushing yards, 241 receiving yards) from scrimmage and 10 touchdowns.