Nothing is pushing it out
Why Does the Water Stay in the Bucket?
Water in a swung bucket keeps travelling straight while the bucket curves round and blocks it.
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Step 1 of 5
Nothing pushes the water outward.
It feels as though something presses the water into the base, but no such outward force exists. Only the bucket ever pushes anything.
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Step 2 of 5
Anything moving keeps going straight.
Left alone, moving water travels in a straight line at a steady speed. Curving requires something to actively push it sideways.
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Step 3 of 5
The bucket keeps getting in the way.
As you swing, the base curves into the water's straight path and shoves it inward, forcing it round the circle with you.
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Step 4 of 5
At the top, gravity does that job.
Overhead, the water still wants to travel straight on. Gravity provides the inward push, so the bucket does not even need to press.
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Step 5 of 5
Too slow and it falls out.
If you swing gently, gravity pulls the water down faster than the bucket moves away, and the whole trick fails at the top of the arc. On an arm's length of string the slowest that works is about one turn every two seconds.
This is exactly why a rollercoaster loop works: the track pushes you inward, and at the top gravity supplies most of the push.
The short version
Nothing flings the water outward: it never turns, and the bucket keeps curving into its straight path and shoving it inward. At the top gravity does that shoving — so swing slower than a turn every two seconds and it spills.
Try it yourself
Swing a bucket with a little water in a fast vertical circle outdoors. Slow down gradually and you will find exactly where it fails.