My Hero Academia has more than a few rivalries, mostly between the students of U.A. Unsurprisingly, the hot-headed Katsuki Bakugo is the quickest to form them, and though his rivalry with Izuku is integral to his character arc, that's his only one. He's also long-held a similar dynamic with Shoto Todoroki.
Bakugo quickly realized that the icy Todoroki was another obstacle to overcome on his quest to become the new #1 hero, and a few scenes in My Hero Academia make it clear that Bakugo takes this rivalry very seriously. However, the two boys also know how to cooperate, and perhaps their relationship is healthier than it might seem at first glance.
When Todoroki Lost To Bakugo In Their Sports Festival Match
The U.A. sports festival was packed with intense matches, and it was here the Todoroki/Bakugo rivalry took root. Until then, Bakugo merely had Todoroki on his radar as a potential rival, and once the two boys fought in the final match, the rivalry took shape at last. Todoroki had broken through his mental barriers to fight Izuku at full strength, but he held back against Bakugo, and the latter saw this as a huge snub, fueling his rage. Bakugo also felt cheated, having won before Todoroki could use his full strength, and this soured his view of the supposedly unbeatable Todoroki.
When Bakugo Swore to Take Todoroki Down
Despite Bakugo's bitterness about Todoroki's lackluster performance, he made it clear that he was not yet done with this rivalry. During the final exams arc, when the students faced their teachers in pairs, Bakugo got in Todoroki's face and declared that he would one day take him down, just the way he would someday take Izuku down.
This makes it clear that Bakugo is the instigator of this rivalry, and Todoroki is on the receiving end. It also proves that Bakugo has faith in Todoroki's power, even if he mocks Todoroki for holding back during their match.
When Bakugo Distanced Himself From Todoroki During Forest Training
After the sports festival wrapped up, Classes 1-A and 1-B traveled to a forest compound to train with the Wild, Wild Pussycats, and Todoroki and Bakugo had an uneasy relationship during this time. Bakugo declared that they were rivals, but he looked down on Todoroki for being a quitter during the tournament, and Todoroki hardly responded to Bakugo's taunts. At one point, they were assigned to undergo the test of courage together, but Bakugo couldn't stand doing that with Todoroki and wanted a new partner.
He didn't get one, so the two boys reluctantly wandered into the forest together, where Class 1-B's students tried to scare them. While Bakugo tried to ignore Todoroki here, sooner or later, Todoroki proves himself worthy of Bakugo's attention.
When Todoroki & Bakugo Completed the Remedial Course Together
Due to their arrogance and uncooperative ways, both Bakugo and Todoroki failed the provisional hero license exam, and they had to take a remedial course, along with a handful of other students. During this time, Bakugo and Todoroki learned to cooperate better with others and tap into their gentler sides. They also learned to trust and respect one another a little more as they cooperated to entertain a class of children.
More so, Bakugo casually mentioned how his parents used violence in his upbringing, but then he reflected on Todoroki's difficult childhood and realized that he might have gone too far. Bakugo starts seeing Todoroki as a person, complete with emotions and personal baggage.
When Todoroki & Bakugo Gained Their Provisional Licenses
Todoroki and Bakugo caught up with the rest of 1-A and obtained their Provisional Hero licenses eventually. Both put those licenses to work right away, too -- fearlessly confronting powerful villains in the streets with their combined might. The police and Pro Heroes were astonished to see just what they were capable of so soon after earning their licenses. Todoroki and Bakugo had become open rivals again, each striving to outdo the other, but in a better way than last time.
Earlier, the boys had been bitterly butting heads in a juvenile clash of wills. Now they've become fierce but respectful rivals and teammates, similar to Bakugo and Izuku's dynamic after Bakugo learned the truth about All For One Quirk.