The Apple Watch usually illuminates the display for 15 seconds. This is because this is the only way to effectively save battery capacity.
However, Apple offers an option in the settings where the screen stays on longer than the default 15 seconds.
You can find out where to find this feature on your Apple Watch here:
1. Open the settings on your Apple Watch.
2. Navigate further to "General" and then to "Activate display".
3. Under the entry "Tap" you can now select a time span between 15 and 70 seconds.
Selects 70 seconds here.
This sets the maximum time for the Apple Watch screen. From now on, the Apple Watch remains switched on for 70 seconds as soon as you tap on the screen.
Note that if wrist recognition is turned on additionally, the display will turn off as expected when you rotate your wrist.