Even though you've shallow copied the cart items array you are still mutating the individual items with the post increment.

const increaseHandler = (x) => {
    let newCart = [...props.cartItems];

    let exist = newCart.find((item) => item.id === x.id);
    if (exist) {
        exist.quantity++; // <-- mutation!!
    } else {
        exist = {
            ...props.cartItems,
            quantity: 1,
        };
    }
    newCart.push(exist); // <-- adds duplicate items
    props.setCartItems(newCart);
};

You still need to create a new item reference, shallow copying the properties and updating the quantity property. You should only push new elements into the cart array.

const increaseHandler = (x) => {
    const newCart = [...props.cartItems];

    let exist = newCart.find((item) => item.id === x.id);
    if (exist) {
        exist = {
            ...exist,
            quantity: exist.quantity + 1;
        };
    } else {
        exist = {
            ...props.cartItems,
            quantity: 1,
        };
        newCart.push(exist);
    }
    
    props.setCartItems(newCart);
};

It's more common to map the previous state to the next state, and I suggest using a functional state update to ensure correctly updating from the previous state versus any state closed over in callback scope (possibly stale).

const increaseHandler = (x) => {
  props.setCartItems(items => {
    const inCart = items.some((item) => item.id === x.id);
    if (inCart) {
      return items.map((item) => item.id === x.id 
        ? {
          ...item,
          quantity: item.quantity + 1,
        } 
        : item);
    }
    return items.concat({
      ...props.cartItems,
      quantity: 1,
    });
  });
};

You can update by using map:

const increaseHandler = (x) => {
  props.setCartItems((preState) =>
    preState.map((item) => (item.id === x.id ? { ...item, quantity: item.quantity + 1 } : item)),
  );
};

You're setting the entire cartItems array into the single exist object you've created. So you're adding the entire cart to exist and then setting exist.quantity to be 1, and pushing that whole thing into newCart and setting it into state.

Try spreading exist rather than props.cartItems in your else block.

Based on your images and nomenclature, is this the FreeCodeCamp/Wiebenfalk tutorial on react & TS? If so, I did the same tutorial, and here's my add to cart function:

const handleAddToCart = (clickedItem: CartItemType) => {
setCartItems((prevState) => {
  // check if item is in cart by looking for id of clicked item in array of cartItems already in state
  const isItemInCart = prevState.find((item) => item.id === clickedItem.id);
  if (isItemInCart) {
    // search for item id matching clicked item; increment that amount
    return prevState.map((item) =>
      item.id === clickedItem.id
        ? { ...item, amount: item.amount + 1 }
        : item
    );
  }
  return [...prevState, { ...clickedItem, amount: 1 }];
});
};