You can easily turn off Snapchat story notifications by following these steps: Open Snapchat and tap your profile icon → gear icon (Settings) → scroll to Notifications → tap Story Notifications and toggle it off. This will stop alerts whenever your friends post stories.
To turn off Snapchat story notifications: open Snapchat → Profile → Settings (gear icon) → Notifications → Story Notifications, then toggle it off. This stops alerts whenever friends post stories.