Westeros is the length of South America (3,000 miles from the Wall to the Summer Sea). The relationship between Westeros and Essos in size is apparently akin to that between South America and Asia, which would roughly make Essos between two and three times the size of Westeros. It's not the case that Essos should be so huge that Westeros looks like a little stick of land off to one side (a mistake I've seen some older speculative world maps make).

We also only have hard info for the sizes of the Free Cities region and Slaver's Bay parts of Essos, which are still basically the western end of the continent. The Dothraki Sea, Qarth, Red Waste, Jade Sea, Shadow Lands, Asshai, Yi Ti, Jhogos Nai etc are all completely unknown in their relative dimensions, but the gist of it is that they extend off to the east for thousands upon thousands of miles. Hopefully we'll see more of this on the official maps (though if they only extend to Asshai, it may be that even those won't show the full extent of Essos, if the unexplored Shadow Lands extend even further east).

However, I agree that Westeros' depiction in the books is a bit odd. It's by far the biggest and most cohesive civilisation in the world (that we know of), not far off the size of the Valyrian Freehold (at least that we know of) in size. It seems to get fairly short shrift when it's power should be very considerable.

ETA: I find the shape of Essos as it's been made out on the newer maps to be aesthetically unpleasing. It really feels like Essos should be a bulky continent which extends north as well as east, but the info we have suggests that Essos extends in a south-easterly direction, which makes the continent fairly narrow around Slaver's Bay and the Dothraki Sea. There's still a fair bit of land in there, but certainly the Dothraki Sea no longer feels like the endless Mongolian/Siberian steppes that the novel seemed to indicate was the case.