In the UK, to describe someone's general loyalty to and support for a sports team, we usually use the word "support". It may be that, in the US, your sentences mean what we in the UK would say as "They support the LA Lakers". , and hope that they win or succeed.
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MUSCOVITE"To support a team" sounds like a formal (not idiomatic at all :-) expression to me...In the UK it is normal at all levels of speech.
MUSCOVITEOn the othe hand, can we say that to "root for (a team)" and to cheer for (a team)" are 100% informal (maybe even slangy?) languageIn the UK, when used in the UK sense