Rückfrage: Wo würdest du das Verb setzen - und warum?

Mir kommt jedenfalls an deinem Beispielsatz absolut nichts merkwürdig vor.

The verb is at the end because the time indicator "In the early 20th century" has been placed at the front and "relations between Russia and the United States" is the subject.

English, since it does not use case markers, is much more dependent on word order than German. As a result, Subject-Verb word order is crucial. The verb follows the subject.

The author, however, wished to emphasize the time frame, so moved "In the early 20th century" to the front of the sentence. That shifted the subject and verb. Unlike German, English does not keep the verb in second position; it keeps the verb after the subject.

Other possibilities:
"Relations between Russia and the United States worsened in the early 20th century." - neutral word order
"In the early 20th century, relations worsened between Russia and the United States." - slight change in meaning or emphasis; "between Russia and the United States" is no longer part of the subject but modifying the verb; the essential meaning is not changed: things got worse.