New York’s primaries on Thursday will be a day of reckoning for a group of Democratic state senators who in recent years had formed a coalition and sided with Republicans, giving control of their chamber to the GOP.

The eight members of the Independent Democratic Conference, which was founded in 2011 and disbanded in April 2018, are facing primary challenges from opponents who are in some cases touting their own progressive bona fides, while pledging to stick with mainline Democrats.

Though...