"demonstrate the link" is correct and usable in written English.
You can use it to explain how two concepts, ideas, events, etc. are related. For example, "The author of this article attempts to demonstrate the link between the rise in tension in the Middle East and recent economic downturns in the region."
He uses the Suez Canal experiment to demonstrate the link between trade and income, which he then uses to support reductions in trade barriers.
Despite that clarity, a bitter fight broke out between Montagnier and Gallo over who should receive credit for discovering the Aids virus (Gallo had done much to demonstrate the link between HIV and Aids).
The committee added that the Department for Communities and Local Government (DCLG) must do more to demonstrate the link between devolution and economic growth, as repeatedly trumpeted by the former chancellor George Osborne.
Removing the boxes doesn't mean we won't know this is what is happening - talk to any food bank volunteer, and they will tell you that bedroom tax, sanctions and benefit delays and the like are having a huge impact on food bank use - it just makes it harder to demonstrate the link in hard statistical terms.
However many attempts to demonstrate the link between symptoms and CD failed.
These data demonstrate the link between ahr2 signalling and transcriptional expression of Ugt genes.
The decision to invite climate scientists as witnesses to the Philippines investigation is seen as a significant opportunity to demonstrate the links between climate change and extreme weather.
Two examples in this section demonstrate the links between public health provision, income disparities, and infectious disease.
Several studies have demonstrated the link between auto use, suburban living and obesity.
There is clear evidence demonstrating the link between the illicit drug trade and an increase in levels of violence, conflict, crime and corruption and decreased citizen security.
Cohen-Solal subtly demonstrates the link between Rothko's three outsider statuses (artist, immigrant, and Jew), his color-block canvases, and his essential Americanness.
