Turn right on Broadway into/onto Caxton Street. It's understood what that means.
---------> 1234 Caxton
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R/R
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But, the OP has a point. Sometimes, in conversation, it can mean "immediately" or "quickly."
For example, someone is asking for directions to 1234 Caxton Street. I can give directions up to driving up Broadway Street. Suppose Caxton Street begins on Broadway and is 100 feet from the railroad.
Then I might say:
Caxton is very close to the railroad.
After crossing the railroad, you want to turn right off Broadway into Caxton Street.
--lotus