Pourquoi pas Je soutiens son équipe? Car supporter apporte un sens négatif en français,je suppose?
Freeform Writing Exercise A1
je soutiens son équipe
je supporte son équipe
were accepted as alternatives and are synonymous in this context.
Soutenir and supporter can both mean "to support" and, in this respect, are close synonyms. Yes, supporter can also have a negative meaning but it isn't restricted to this. Supporter can also mean a more general support, like, e.g., the foundation that supports the weight of the building.
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