JavaScript ES6, 119 bytes
F=s=>(C=o=>--a.length?C(a.reduce((p,c,i)=>c+p.slice((a[i-1]=p.slice(0,c.length)).length)))+` `+o:o)(a=(s+` `).split` `) Here it is ungolfed and in ES5 with comments explaining how it works:
function F(s) { var arr = (s+'\n').split('\n'); // Create an array of words and append an empty member return (function C(output) { return --aarr.length ? // Remove the last item from the array C(aarr.reduce(function(p,c,i) { // If the array still has length reduce it to a string and recurse var intersection = (a[iarr[i-1] = p.slice(0, c.length)) // Overwrite the previous word with the part that intersects the current word return c + p.slice(intersection.length) // Add the part of the previous word that doesn't intersect to the current value })) + '\n' + output : output // Add the last level of recursions output on to the end of this })(a=(s+'\n').split('\n')).trim(arr) // Create an array of words and add an additional member; } input.addEventListener('input', updateOutput, false); function updateOutput() { var oldLength = input.value.length; var start = this.selectionStart; var end = this.selectionEnd; input.value = input.value.split(/ +/).join('\n'); var newLength = input.value.length; input.setSelectionRange(start, end + (newLength - oldLength)); output.value = F(input.value).trim(); } updateOutput();textarea { width: 50%; box-sizing: border-box; resize: none; float: left; height: 10em; } label { width: 50%; float: left; }<p>Type in the input box below, spaces are automatically converted to newlines and the output updates as you type</p> <label for="input">Input</label> <label for="output">Output</label> <textarea id="input"> Type inside me :) </textarea> <textarea id="output" disabled> </textarea>