How can small cap index funds grow in value if the ones that grow are turned over?

How can small cap index funds grow in value if the ones that grow are turned over?

Well they aren't turned over they are sold. The money from the sales is then used to buy either additional shares of companies in the fund, or used to buy shares of companies that are new to the fund.

The definition you quote:

A small cap is generally a company with a market capitalization of between $300 million and $2 billion.

It is not set in stone. Different groups of people use different definitions, even when they are talking about the same country. Those definitions also change over time. That wasn't the definition 10 years ago.

Yes there will be stocks that quickly move through the range on their way to being huge. But others will spend many years or even decades in the small-size, or the mid-size definitions.

It sounds like this index fund just remains "small" forever due to keeping the basket within the bounds of the definition of a small cap company.

This can be a problem. Some active funds run into problems when they get so large that the amount of money they need to invest in their next pick moves the market price so much that they destroy the opportunity. That causes them to change their approach, or even to close the fund to new investors.