After leading UCLA in blocks as a freshman reserve, he was a member of the United States national team that won the gold medal at the FIBA Under-19 World Championship in 2015.

He started for the Bruins as a sophomore and junior, leading the team in rebounds while also continuing to be their top shot blocker.

[7] The following summer, Welsh performed well with the Los Angeles Rockfish, the longest ongoing high school all-star program in Southern California.

[8] He emerged as one of the top centers in California as a senior, and was selected for the 2014 McDonald's All-American Game,[5] a rarity for a player who did not receive national recognition as a junior.

[16] He had six rebounds and four blocked shots in 22 minutes in an upset victory over SMU in UCLA's opening game of the 2015 NCAA tournament.

Playing the final minutes of the game in place of Parker, he made a key block with 34 seconds remaining and UCLA down by four.

[19][20] In the 2015–16 season opener, he had his first collegiate double-double with 12 points, 10 rebounds and a career-high five blocks in an 84–81 overtime loss to Monmouth.

[23][24] On January 9, 2016, he had a season-high 16 rebounds in an 81–74 win over Arizona State,[25] which helped UCLA avoid falling to 1–3 in the Pac-12 for the first time in almost 20 years.

[28] Welsh ended the season with an 11.2 scoring average, and led UCLA in field goal percentage (59.0), rebounding (8.5) and blocked shots (1.0).

[31] On February 19, 2017, he scored 16 points and had a season-high 16 rebounds in a 102–70 win over their crosstown rivals, USC, ending a four-game losing streak against the Trojans.

[33] In the Bruins' regular-season finale, he had 13 points and 11 rebounds and also made his first career three-point field goal in a 77–68 win over Washington State.

[35][38] Over the summer, Welsh worked on his three-point shooting based on feedback from his NBA workout with the Oklahoma City Thunder.

He was forced him to wear a protective mask after halftime, which affected his shooting as UCLA lost 107–99 in double overtime.

[43] On February 15, he played his first game without a mask since January and logged a career-high 17 rebounds along with 15 points in a 75–68 victory over Oregon State.

[56] Nuggets assistant coach Wes Unseld Jr. was impressed early by Welsh's constant talking—calling out coverages and directing teammates on the court—in spite of his youth and inexperience.

[57] Welsh made his NBA debut on October 17, 2018, against the Los Angeles Clippers, seeing under one minute of game action and scoring no points.

[67] Welsh tried out for the United States national team in the 2014 FIBA Americas Under-18 Championship, but was one of the final three roster cuts.