First of all, it is very important to note that quantum entanglement is not a spooky action-at-a-distance as Einstein once called it! It is a strong correlation of measurements that is stronger than any classical correlation could ever be. This has been experimentally verified by the so-called violation of Bell's inequalities.
Second, quantum entanglement has been successfully demonstrated for distance of some 100 km, as the Wikipedia article on quantum teleportation states (references in Journales such as Nature are given there)
Zeilinger’s experiments on the distribution of entanglement over large distances began with both free-space and fiber-based quantum communication and teleportation between laboratories located on the different sides of the river Danube. This was then extended to larger distances across the city of Vienna and most recently over 144 km between two Canary Islands, resulting in a successful demonstration that quantum communication with satellites is feasible. His dream was to bounce entangled light off of satellites in orbit.[1] This was achieved during an experiment at the Italian Matera Laser Ranging Observatory.[4]
Admittedly, that is not a demonstration at infinite distance, but then, nothing has ever been demonstrated at that length scale and we would never be able to do and to demand such a demonstration is unreasonable. But please not the different scales involved here! Usually, we think of quantum mechanics to govern the microscopic world, involving length scales of under a micrometer, i.e. $10^{-6} m$. Achieving quantum effects on a length scale of $100 km = 10^5 m$ means you span 11 orders of magnitude. You can jokingly call this "infinite for all practical purposes".
The misunderstanding is that nothing changes in the moment of your measurement. There is a classical analogue that is wrong on the details but emphasizes the main idea: Suppose you take two playing cards, one is a king and the other a queen. You shuffle them and let your friend pick one at random while you keep the other one. Now, you have the king with probability $1/2$ and the queen also with probability 1/2, but you don't know which one you have without looking. You also don't know which one your friend has. But as soon as you look at your card, you immediately know which one your friend has: If you got the king, he must have got the queen. There is no spooky action at a distance that somehow changes what the card of your friend looks like!
In quantum mechanics, it is admittedly a bit more complex, because not only would you not know what card you have without looking, it isn't even determined until you look. It is, however, possible to go through the math and the mechanisms and come up with a satisfactory explanation that does not involve faster-than-light communication and processes.
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