TLDR: In Singapore, Primary School touches on the well-known ancient civilizations. History courses in Secondary School focus mainly on Singapore, mainly adding in European history in the context of colonization or major world events (Nazism, communism, etc.) Very little is taught about medieval Europe or the middle East.

I'm Singaporean. In Singapore, much of our primary school social studies education from Ages 7 to 12, is intended to indoctrinate young students into better understanding the multicultural society that is Singapore. You can examine the Ministry of Education's syllabus for Primary School Social Studies here.. Just for fun, here is the syllabus for the 'Character and Citizenship Education'.

As mentioned in the syllabus, Primary Five is meant to provide an overview of several ancient civilizations and their achievements:

The Sumerians - Writing, Irrigation

The Egyptians - Sun-dial, The Pyramids of Giza

The Greeks - The Olympics, The Temple of Artemis

The Romans - Roman Aqueduct, The Colosseum

However, there is an entire section of the syllabus devoted to "[how] the legacies of China and India influence our lives today", which discusses common topics like the Huang He, ways of life in Ancient and modern China, Chinese technological inventions and the same things for the Indus Valley civilizations.

Then, in Primary Six, the social studies curriculum reorientates itself around Southeast Asia more specifically. We learn about famous kingdoms in SEA, famous UNESCO sites, present day industries, etc.

After twelve year olds take the Primary School Leaving Examination (PSLE) and are separated into different education tracks, the syllabuses begin to vary. 'Express' and 'Normal (Academic)' track students study history (who, what, when, where) focusing mainly on Asia. Lower Secondary focuses on Singaporean history. For example:

Singapore and Asian Trade situation before the 14th century

Rise of Temasek in the 14th century

How was life different for the various people in colonial Singapore before World War Two?

Singapore's connections with Melaka (15th to 16th centuries)

Singapore's connections with the Johor-Riau Sultanate (16th to early 19th centuries)

Singapore's connections with the Portuguese and Dutch (16th to early 19th centuries)

British failure to defend Singapore

Life under the Japanese

Views of the British as rulers of Singapore

British superiority

On the other hand, Upper Secondary focuses more on Europe, but only briefly in the context of colonization, WW2, and the Cold War. For example:

Rise of authoritarian regimes and their impact in the Interwar period

Reasons for the rise of Stalin in Russia

Impact of Stalin's rule in Russia

Reasons for the rise of Hitler in Germany

Weaknesses of the Weimar government

Hitler's consolidation of power, one-party rule

Weaknesses of the League of Nations

Hitler's aggressive foreign policy

Japan's worsening relations with the USA

Over-extension of Hitler's army on many fronts

US-Soviet mistrust post-war: breakdown of wartime alliances, division of Europe after WW2, American containment policy, Soviet response, military responses

Tensions between USA and Castro: Castro's policies and American responses

Role of external powers in the conflict: USA and USSR (Kennedy and Khrushchev)

We also touch on Dutch colonies in Indonesia, British Malaya, and the French failure in Vietnam.

Social studies courses for all tracks, including Normal (Technical), are intended to promote racial and political harmony, essentially functioning as a basic civics course.

I know it isn't about China, but if you're interested in learning more, check out the Ministry of Education's syllabuses here.

I will talk about Indian schools. There are multiple boards which can be broadly classified into state boards and central boards. While the curriculum is similar state boards have a little more of that particular state history so treatment of other nations history might differ a bit. But Central boards are nationwide and hence can be a better generalization of what the students in the country are expected to know. Also, one difference from many other countries we have is that students specialize in grade 11 and 12th. So not everyone studies history there. Hence class 6th-10th is what everyone studies. However 9th-10th is often repeat of 6th-8th in detail.

History starts with pre-historic period and quickly moves on to early civilization. Indus valley, Mesopotamian, Egyptian and Chinese civilization are taught. Then Early Vedic period, rise of Buddhism and Jainism, Magadha empire with passing mention of Alexander. Maurya Dynasty in detail. This includes greatest Ancient Indian ruler Ashoka. Decline of Jainism and Buddhism. Shakas and Satraps, Gupta Dynasty, Southern Kingdoms, Other Hindu Kingdoms until around 800 AD with detailed mention of Harsha. All this is part of ancient civilization and is taught in one year. In some of the state boards, especially southern there is more focus on southern kingdoms and their conquests.

2nd year, 7th grade, is for medieval. Starts somewhere around invasions of Mahmud of Gazhni and Muhammad Gori. Formation of the slave dynasty followed by the rise and fall of various Delhi sultanates. Then the Mughal dynasty is covered in great detail. Also covers the South Indian Vijayangara empire. Around 17th century covers various Indian kingdoms and conflict with Mughal dynasty, including Shivaji. Akbar is the greatest medieval figure. This goes on until fall of mughal dynasty.

8th grade is for modern history. With brief mention of early European starts aroudn fall of Mughal dynasty and rise of European powers. British conquests, first war of independence in 1857 in detail, covers until first world war. Then the Gandhi era after first world war. Non cooepration movement, civil disobedience movement, quit India around world war 2, independence and then little bit after that.

Only non India history taught in ICSE is ancient civilizations, some invasions during ancient time, greek satraps during early CE, the two world wars and evils of imperialism. Most of the time it is India centric.

So speaking from my experience as a social studies teacher who taught 9th grade, I can attest to the fact that this isn't really the case anymore. Key word their being "anymore". In the last decade, especially since the introduction of common core, the Social Studies framework in my state dictates teachers to focus on China, Egypt, Mesopotamia, and India during the river valley age, and then maintains focus on these areas when discussing the rise of empires such as Greece, Persia, China, Egypt, etc.

Typically the focus shifts to Greece and Rome, and then into Europe, but the ancient portion of the US history class misses a large chunk of ancient European history that I think would greatly improve student understanding, especially with regards to early Greek peoples, and the Etruscans. In fact my usual history textbook that my students read from only has 2 pages on Ancient Greece and nothing aside from a brief mention of Etruscans leading into Rome.

Source: New York State Common Core Framework

http://www.nygeographicalliance.org/sites/default/files/ss-framework-9-12.pdf

Can't find an online version of my textbook, but will update if I do.

The premise of your question is flawed, and really I'm not sure where you're getting your information/recollection.

Though social studies does not fall under the Common Core standards, the US education system as per the vast majority (though I can't speak necessarily for the totality) of the states curricula at both the middle and high school level World history courses (typically required to graduate at both levels) does indeed focus on all four of the so-called "four culture hearths" of humanity - The Nile River Valley, Mesopotamia, the Indus River Valley, and the Yellow River Valley. Moreover, throughout the course of the year, all modern textbooks and curricula cycle back though those culture hearths and the subsequent civilization there and beyond. The focus today is far more on cultural interaction, comparison, and the flow of technologies, information, ideas, and diseases from one to the others than you question makes it out.

For instance, here is the first two chapters from the Gr. 10 WH textbook my school uses, as aligned with the California State Social Studies Curriculum:

Ch.1 - The Rise of Civilization: 1.1 Early Humans, 1.2 The Neolithic Revolution, 1.3 Mesopotamia

Ch. 2 - The Spread of Civilization: 2.1 The Rise of Egypt, 2.2 The Peoples of the Eastern Mediterranean, 2.3 The Indus Valley Civilization, 2.4 The Rise of China, 2.5 Civilizations in the Americas

The then goes into the Near Ancient East, Ancient Greece, India's First Empire, First Chinese Empires, The Romans, Byzantines and Emerging Europe, Islam and the Arab Empire, Medieval Europe, Civs of East Asia, the Crusades, Kingdoms and States of Medieval Africa, Pre-Colombian Amercias, the Renaissance, the Reformation, The Age of Exploration, European Absolutism, Central Asian Empires, Post-Mongol East Asia, Enlightenment and Revolutions, French Rev and Napoleon... etc...

It's still rather Western/Euro-centric... but not nearly as exclusionary as you imply.

Are you sure you're not misremembering your Euro History class as a World History class?

In my high school (15 yrs ago), the history class is based on two sets of textbooks: 1) world history and 2) Chinese history. You have to take the mandatory Chinese literature class at the same time, which heavily focused on about 200 classic Chinese literature articles dated from 1000bc to early 20th century(and one have to remember about 80 of them to pass exams). To understand those, you have to learn all the context and background. That also teaches students a great amount of history.