How to turn off horizontal split screen on iOS 14 on an iPhone?
Unlike iPadOS (variant of iOS, renamed to reflect features specific to iPad, such as the ability to view multiple running apps at once), iOS has no ability to view two or more running apps in a split screen mode.
What you are encountering is an iOS (running on iPhone) feature called Reachability. Reachability is a feature available on iPhones with an iPhone 6–sized screen or larger. The rationale is to make single-handed use of larger screens easier by temporarily shifting the on screen elements to make them easily accessible.
You can learn more about Reachability in the official iPhone user guide.
Reachability is enabled by default in iOS. On iPhones with a home button, Reachability can be invoked by double tapping on the Home button. On iPhones without a home button (i.e. those that use Face ID), Reachability can be invoked by swiping down on the bottom edge of the screen.
Reachability is available on the following iPhone with Home button:

iPhone 6

iPhone 6 Plus

iPhone 6s

iPhone 6s Plus

iPhone 7

iPhone 7 Plus

iPhone 8

iPhone 8 Plus

iPhone SE 2nd gen


and on the following iPhone which come without Home button/with Face ID:

iPhone X

iPhone XR

iPhone XS

iPhone XS Max

iPhone 11

iPhone 11 Pro

iPhone 11 Pro Max

iPhone 12

iPhone 12 mini

iPhone 12 Pro

iPhone 12 Pro Max


You can move out of Reachability mode by double tapping the Home button or swiping up on the bottom edge.
Reachability mode is not persistent after invoking and the iPhone also automatically moves out of Reachability mode after a little while once you start interacting with the on screen elements.
If you do not wish to use Reachability feature, you can disable it on your iPhone by going to Settings app → Accessibility → Touch and turning the switch corresponding to Reachability off.
 

To answer your specific questions:

How to turn off horizontal split screen on iOS 14 on an iPhone

By turning off Reachability from within Accessibility settings.

how do I make sure the screen does not split?

By turning off Reachability.

Which gesture even causes the split?

Swipe down on the bottom edge on iPhone without Home button/with Face ID.