Is there are way to make a child DIV's width wider than the parent DIV using CSS?

Here's a generic solution that keeps the child element in the document flow:

.child {
  width: 100vw;
  position: relative;
  left: calc(-50vw + 50%);
}

We set the width of the child element to fill the entire viewport width, then we make it meet the edge of the screen by moving it to the left by a distance of half the viewport, minus 50% of the parent element's width.

Demo:

* {
  box-sizing: border-box;
}

body {
  margin: 0;
  overflow-x: hidden;
}

.parent {
  max-width: 400px;
  margin: 0 auto;
  padding: 1rem;
  position: relative;
  background-color: darkgrey;
}

.child {
  width: 100vw;
  position: relative;
  left: calc(-50vw + 50%);

  height: 100px;
  border: 3px solid red;
  background-color: lightgrey;
}

<div class="parent">
  Pre
  <div class="child">Child</div>
  Post
</div>

Browser support for vw and for calc() can generally be seen as IE9 and newer.

Note: This assumes the box model is set to border-box. Without border-box, you would also have to subtract paddings and borders, making this solution a mess.

Note: It is encouraged to hide horizontal overflow of your scrolling container, as certain browsers may choose to display a horizontal scrollbar despite there being no overflow.

Use absolute positioning

.child-div {
    position:absolute;
    left:0;
    right:0;
}

I've searched far and wide for a solution to this problem for a long time. Ideally we want to have the child greater than the parent, but without knowing the constraints of the parent in advance.

And I finally found a brilliant generic answer here. Copying it verbatim:

The idea here is: push the container to the exact middle of the browser window with left: 50%;, then pull it back to the left edge with negative -50vw margin.

.child-div {
  width: 100vw;
  position: relative;
  left: 50%;
  right: 50%;
  margin-left: -50vw;
  margin-right: -50vw;
}