Inflationary theory allows for a solution to the problem (along with several others such as the flatness problem) by positing a short 10 − 32 second period of exponential expansion (dubbed "inflation") within the first minute or so of the history of the universe. During inflation, the universe would have increased in size by an enormous factor.
If correct, inflation solves the horizon problem by suggesting that prior to the inflationary period the entire universe was causally connected, and it was during this period that the physical properties evened out. Inflation then expanded it rapidly, freezing in these properties all over the sky; at this point the universe would be forced to be almost perfectly homogeneous, as the information needed to change it from that state was no longer causally connected. In the modern era distant areas in the sky appear to be unconnected causally, but in fact were much closer together in the past.
Note that the solution involves almost (99.99999999% - it's more but I'm only allocating so much space for 9's) all of the stuff that supposedly became the universe moving way, way, way, WAY faster than light.
It takes light minutes just to get from here to the sun. Now in 10 to the -32 second, even light can't go all that far. Yet the universe is supposed to have expanded to like 90% of its present size during that infinitesimal fraction of a second. They don't like to specify how much, mind you, lest anyone who doesn't already should begin to see how bogus the "theory" really is.
See, here's how a typical example words it:
According to astronomers, the universe grew from submicroscopic to astronomical size in its first trillionth of a second.
No mention of just how big - just the vague "astronomical size". Even so, light would be left in the dust by this stuff.
Is there a workaround? Ha! You kidding? "The laws of physics didn't apply then." But there's still a problem, because all this unspecified stuff has to get slowed down under the speed of light before the "law" kicks in. Nothing can travel at the speed of light, so anything traveling faster when "the law kicks in" will be trapped forever traveling faster than light. "The law" prevents such things from decelerating to sublight speeds.