[4] As a student at the University of Southern California, Morozov started competing collegiately and studying economics in pursuit of an undergraduate degree in 2010.

In the 50-metre freestyle, Morozov defeated 2012 Olympic champion in the event, Florent Manaudou of France, as well as Anthony Ervin of the United States, and set the national record with a time of 20.55 seconds.

[24] Later in the same finals session, he split a 20.44 for the second leg of the 4×50-metre mixed freestyle relay to help win the gold medal in 1:29.73.

[26] Approximately 15 minutes later, Morozov placed sixth in the final of the 100-metre individual medley with a 52.83, which was 0.99 seconds slower than the gold medalist in the event Michael Andrew of the United States.

[28] In the morning of day five, Morozov placed 18th in the prelims heats of the 100-metre freestyle with a 47.82 and did not advance to the semifinals stage of competition.

[15][32] During the 2017 FINA Swimming World Cup stop in Berlin in August, Morozov swam a 45.23 and broke the 100-metre freestyle Russian national record from eight years prior set at 45.36 by Yevgeny Lagunov.

[8][33] At the 2017 European Short Course Swimming Championships in Copenhagen, Denmark in December, Morozov set a new Russian national record in the 50-metre freestyle twice, swimming a time of 20.31 seconds and winning the gold medal in the event final after first lowering the national record to 20.45 seconds in the semifinals.

[36][37] In early November at the World Cup stop in Tokyo, Morozov swam 50.26 seconds in the 100-metre individual medley again, tying his own pre-existing records.

[43][44] At the long course 2019 FINA Swimming World Cup series stop in Singapore, Morozov lowered the Russian record in the 50-metre freestyle by almost two tenths of a second to 21.27 seconds and tied Bruno Fratus of Brazil as the tenth fastest performer in the race to that point in time.

[57] For the time spanning 17 December 2020 through 16 December 2022, Russians, including Morozov, were not allowed to compete using their country name, anthem, nor flag at World Championships, including Olympic Games, due to a ban originating as a reaction to widespread doping in Russia and enacted by the Court of Arbitration for Sport.

[70] Stop number two of the World Cup series, held in Budapest, Hungary, Morozov started his competition on day one, swimming a 21.51 in the 50-metre freestyle prelims in the morning and advancing to the final in the evening.

[72][73] Day two of competition, Morozov ranked third in the 100-metre freestyle prelims heats at 47.41 seconds and advanced to the final.

[82] Morozov won his first gold medal of the 2021 World Cup circuit in the final of the 50-metre freestyle, finishing first with a time of 20.89.

[83][84] In the morning prelims on day two, Morozov ranked fourth overall with a time of 47.56 in the 100-metre freestyle, just one hundredth of a second behind Kyle Chalmers heading into the final.

[85][86] He cruised to winning a silver medal in a time of 46.31 seconds in the final, finishing only after Kyle Chalmers.

[64]Making good on his promise to try to beat Kyle Chalmers, Morozov ranked third overall in the prelims heats of the 50-metre freestyle with a time of 21.25 seconds that was just one tenth of a second behind Kyle Chalmers, who ranked first in the heats, and both advanced to the final later in the day.

[98] Morozov finished off the podium in the final, coming in fifth-place and thirty-five hundredths of a second behind Kyle Chalmers who won the bronze medal.

[103] The day before the start of competition, an updated entries list revealed Morozov had pulled out of racing the 100-metre individual medley at the Championships.

[105][106] On the second day, Morozov competed in his first individual race, swimming a 21.23 in the prelims heats of the 50-metre freestyle and qualifying for the semifinals.

[107] In the semifinals of the 50-metre freestyle, Morozov lowered his time from the prelims by a quarter of a second, swimming a 20.98 and advancing to the final ranked third.

[118][119] The second day of competition, he led-off the 4×50 metre mixed freestyle relay in the final with a 21.17, helping achieve a bronze medal-winning time of 1:28.97.

[120] In the prelims heats of the 50 metre freestyle on day three, he did not qualify for the semifinals, ranking 17th with a time of 21.54 seconds.

[123] On day five, he split a 20.37 for the freestyle leg of the 4×50-meter medley relay in the final, contributing to a Championships record and gold medal-winning time of 1:30.51.

[124][125] On 21 April 2022, the world governing body for aquatic sports, FINA, banned Morozov and all of his fellow countrymen and countrywomen, as well as all Belarusians, for at least the remainder of the 2022 from any of their competitions or other events.

On April 19th, 2024, Morozov officially announced his retirement from competitive swimming during an interview with Match TV.