The logical answer is in an expanding earth theory - work backwards and you will have a much smaller planet, with all sides of all continents fitting together.
But, the skeptic answers, what causes the earth to expand? The straw man is there has to be more mass falling on the earth from the sky, but there does not seem to be enough by current measurements.
But if the earth's core is hot and this causes the expansion and cooling, it would not only cause separation of plates but it would do so in a manner which just as easily explains their movements.
The challenge of course is to measure the growth - can anyone spare a few satellites?