How to remove the index name from the URL ?

In InstantSearch version 4 we added federated search which means that the uiState might have multiple indexes that need to be stored in the URL, this is why they are now in the URL by default. There is more detail on this here: https://www.algolia.com/doc/guides/building-search-ui/upgrade-guides/js/#routing

However, this can be easily removed by specifying the index you are using, for example:

routing: { 
    stateMapping: instantsearch.stateMappings.singleIndex('your_index_name') 
}

There is also documentation for the routing parameter here:  https://www.algolia.com/doc/api-reference/widgets/instantsearch/js/#widget-param-routing

Here is a working Codesandbox example of this: https://codesandbox.io/s/routing-remove-index-name-we8gw?file=/src/app.js

Magento extension only

To make this change within Magento 2 you will need to change the search options to include this routing change: https://www.algolia.com/doc/integration/magento-2/customize/instant-search-page/?client=php#change-search-options