Do the TV's have HDMI Control turned on? If so, turning one off may cause the other to turn off.

Try this: Turn both TV's on. Unplug the HDMI cable that goes from the splitter to one of the TV's, and then use the remote to turn the other TV off. If the TV that you've disconnected from the splitter stays on, then the problem is caused by the HDMI Control signal being passed from one TV to the other, via the splitter.

See if you can go into the TV's menu system and turn "HDMI Control" off.