Cards.js

import React from 'react';
import {View, Dimensions} from 'react-native';
import Animated, {
  useAnimatedScrollHandler,
  useAnimatedStyle,
  useSharedValue,
} from 'react-native-reanimated';

const BORDER_RADIUS = 10;

const SCREEN_WIDTH = Dimensions.get('screen').width;

const palette = [
  '#FFB3B3',
  '#FFDCB3',
  '#FFF8B3',
  '#E2FFB3',
  '#B3FFDB',
  '#B3DBFF',
  '#B3BBFF',
  '#DEB3FF',
];

const CARD_WIDTH = 260;
const IMAGE_HEIGHT = Math.floor((CARD_WIDTH / 16) * 9);
const CARD_HEIGHT = IMAGE_HEIGHT + 100;
const MARGIN = 16;
const SCALE = 0.85;

const Card = ({index, lastOne, scroll}) => {
  const animated = useAnimatedStyle(() => {
    let value = 1;
    const current = Math.round(scroll.value);

    if (index === current - 1 || index === current + 1) {
      const offset = 2 * Math.abs(scroll.value - current);
      value = SCALE + offset * (1 - SCALE);
    }

    return {
      transform: [{scale: value}],
    };
  });

  return (
    <View
      style={{
        width: CARD_WIDTH,
        height: CARD_HEIGHT,
        marginRight: lastOne ? 0 : MARGIN,
      }}>
      <Animated.View
        style={[
          {
            width: CARD_WIDTH,
            height: CARD_HEIGHT,
            borderRadius: BORDER_RADIUS,
            backgroundColor: '#fff',
            shadowColor: '#000',
            shadowOffset: {
              width: 0,
              height: 4,
            },
            shadowOpacity: 0.1,
            shadowRadius: 4,
          },
          animated,
        ]}>
        <View
          style={{
            width: CARD_WIDTH,
            height: IMAGE_HEIGHT,
            backgroundColor: palette[index],
            borderTopLeftRadius: BORDER_RADIUS,
            borderTopRightRadius: BORDER_RADIUS,
          }}
        />
        <View style={{padding: 16}}></View>
      </Animated.View>
    </View>
  );
};

const Cards = ({items}) => {
  const scroll = useSharedValue(0);
  const handleScroll = useAnimatedScrollHandler(({contentOffset: {x}}) => {
    const current = x / (CARD_WIDTH + MARGIN);
    scroll.value = current;
  });

  return (
    <Animated.ScrollView
      snapToInterval={CARD_WIDTH + MARGIN}
      contentOffset={{
        y: 0,
        x: (CARD_WIDTH + MARGIN) * Math.floor(items.length / 2),
      }}
      decelerationRate="fast"
      horizontal
      showsHorizontalScrollIndicator={false}
      onScroll={handleScroll}
      scrollEventThrottle={16}
      style={{
        top: 0,
        left: 0,
        position: 'absolute',
        paddingVertical: 20 + 16,
      }}
      contentContainerStyle={{
        paddingHorizontal: (SCREEN_WIDTH - CARD_WIDTH) / 2,
      }}>
      {items.map((item, index) => (
        <Card
          index={index}
          scroll={scroll}
          lastOne={index === items.length - 1}
        />
      ))}
    </Animated.ScrollView>
  );
};

export default Cards;

Works great. If it helps anyone else, ScrollView has a property called disableIntervalMomentum that will stop scrolling on each index, regardless of how fast the gesture is. (I noticed on the video and my own testing that it is possible to flick through more than one item per gesture without disableIntervalMomentum={true}, but maybe some people want that.)