World does not collapse after every Yuga.
The collapse of earth & solar system only occurs at end of 1000 Chatur-Yugas (1 chatur yuga = krita + treta + dwapara + kali yuga = 4.32 million years)
And that too is only partial because higher worlds like Satya loka will still exist as long as Brahma lives, whose life-span is 100 years, where each year has 360 days, and each day = 2000 Chatur-Yugas (1000 daytime, 1000 nighttime)
After Kali Yuga ends (4 lakh+ years from now), there will be a period of prolonged rains, and then Krita Yuga will start all over again.
The age of earth can be easily calculated as one-daytime of Brahma = 1000 * 4.32 million, or 4.32 billion years.
Scientific analysis of carbon dating rocks comes pretty close to this value (4.54 ± 0.05 billion years). Over time science might get more refined and arrive at same answer as given in our ancient scriptures.