The Arabs conquered North Africa (which speaks Arab to this day), Persia, Spain (from the 700 to the 1400, a fairly big ammount of time) and they where able, by means of maritime commerce and the spead of Islam, to influence lands as far as Malaysia and the Philipines.

The Turks conquered Constantinople, the capital of one of the most advanced civilizations in Europe, and they employed the new cannon technology to successfully besiege the city in 1453.
The Mughal empire pretty much colonized India in the 1500s, and did it basically with gunpwder. [1]

Xinjiang, Tibet, and Vietnam were occupied by different Chinese dynasties several times.
While it began in the XVII century, China pretty much colonized Taiwan, despite previous Spanish/Dutch.
China had a significant commercial presence in the Indic Ocean and Southeast Asia by the 1400's, and substantion immigration to SE Asia - specially to Malaysia.
Ming China had substantial naval technology, and they have repeatedly won naval battles against the Portuguese, one of the great naval powers of the time [2].
They invented fire-arrows, basically incendiary rockets, frequently used agrains ships. [3]

The Arabs, Turks and Chinese of the 1400-1600 were colonising/conquering other lands, making maritime commerce and exploration, and had gunpowder technology good enough to beat European powers at the time. The Arabs specially had a strong presence in Africa and did as much slaving as the Portuguese.

So in terms of technology, the civilizations in Asia where more or less equivalent to Europe. They did not fail at colonisation, they just had much less to colonise, and less interest to explore. The reasons are:

1- Of the advanced civilizations of the time, Europe was the closer to Americas, a large, a rich continent whose inhabitants did not use metals nor gunpowder as in the rest of Eurasia. Native Americans had no immunity to Old World diseases too.
The Turkish or Chinese could very easily defeat the Incas and colonise portions of the Americas, had it been closer and they knew about it in time to do so. But Europe was closer, they discovered it first, and the countries of Europe stated competing to colonise it. By the time Asians realized the existence of America, it was probably totally colonized.

2- Europe was very interested in commerce with India, specially access to spices, and America was pretty much discovered by accident in an attempt to find a new route to it [4].
The Chinese and Arabs already had easy access to the Indic Ocean Commecial zone, so they had little incentive to expeditions. The Chinese, in particular, could simply grow their own spices, produced their own silk and porcelain and had little interest in foreign products, and even less interest in expading their religions (the Muslims did spread Islam as far as the Philipines and even to China)
The Europeans sufferd from Turkish commercial blockades, and had to contour Africa to have access to Indian Continent, which was a long and dangerous route. So they eventually tried a new route.

tl;dr:
Europe discovered and colonized the Americas because it was nearer and because of the lucrative spice trade. Asia did not because it is more distant and because they could grow their own spices.

[1] https://en.wikipedia.org/wiki/Mughal_Empire
[2] https://en.wikipedia.org/wiki/Battle_of_Tunmen
[3] https://en.wikipedia.org/wiki/Fire_arrow
[4] https://en.wikipedia.org/wiki/Voyages_of_Christopher_Columbu...