Why high tides occur simultaneously on opposite sides of the Earth?
First of all, tides are not as simple as the &quot;two-bulge&quot; simplification. In reality, the diagram shown is misleading. The two bulges appear assuming an ocean of constant depth covers the entire surface of Earth. Clearly that is not the case and in the diagram you can see the continents. Considering the different sizes of the basins and the distinct frictional characteristics in each location, the resulting tidal effect is much more complex. The difference in phase and amplitude is shown here and it clearly shows that the the tide varies for the same longitude. That wouldn't be the case in the simple explanation above.

Source Wikipedia.
Looking at this tidal animation from TPXO is also illustrative.

The simple &quot;two-buldge&quot; explanation would result in a pure two peak daily tide. That is certainly not the case in places like the Gulf of Mexico.
As mentioned in Camilo Rada's answer, the bulges are a consequence of the tidal force. This  apparent force result from the difference in strength in the gravitational field. The result is that Earth's body is stretched toward and away from the center of mass of the Earth-Moon system. The water thus adjust to this difference in geopotential giving rise to the tides.
A more intuitive explanation is given in Project Earth Science: Physical Oceanography

The explanation of the two-bulge tide comes from the fact that the Moon and Earth form a two-body system that rotates about an axis located within Earth.
The bulge of water on the side of Earth that faces the Moon is easily explained. It is due to the gravitational attraction between the Moon and Earth, including the water on Earth. This attraction pulls water toward the Moon and creates a “bulge” on the surface of Earth. The bulge on the other side of Earth is due to inertia. Inertia is the tendency of an object at rest to stay at rest and the tendency of a body in motion to continue its motion in a straight line.
There is an inertial tendency resulting from the rotation of the Earth-Moon system for objects (water among them) to move away from both sides of Earth—the side facing toward the Moon and the side facing away. The model demonstrates that the effect of things moving away from Earth is much greater on the side facing away from the Moon.
Many textbooks and other sources use the concept of “centrifugal force”— which is actually a preconception—to explain the effects of inertia. According to this preconception, there is a force that acts on all objects that are in circular motion, and this force pushes or pulls the object out from the circle. There is no such force. The preconception arises from our own experience with circular motion.

The gravitational forces of Earth, Sun and Moon cause a bulge of water on the nearest side and an equal bulge on the other side. Thus, in this simple scenario, the tide is composed of two bulges of water (four, in fact), traveling around the world as the world spins. When Moon and Sun aligned, their respective bulges add together to form &quot;spring tides&quot; every two weeks. When the Moon and Sun are at right angles, we encounter &quot;neap tides&quot;, as the bulge of the sun adds to the low lunar tide, resulting in higher low tides but lower high tides.
The limitations of this model are:

It cannot explain that there are places without tides, with one daily high, and most with two tidal highs each day.
Tidal height is not maximal at the Equator (and minimal at the poles) as the simplification suggests.
High tide is not associated with the position of the Moon. It occurs at different times of the lunar cycle depending on the location.
If continents are included, the tidal wave would reflect off the continental shelf as it reaches a continent. A tidal wave of almost equal magnitude will be propagating in the opposite direction, which is not observed.
The tidal waves required for this model would have to travel at much faster speeds that are possible in reality.

In reality, the tides instead of running east to west as Earth rotates, tidal waves propagate around in circles around islands, and certain points in the sea, called tidal nodes or amphidromic points. These nodes can be seen in the first figure from this answer.
Thus, the tidal patterns in the ocean are a set of rotating standing waves. These waves have periods that represent the natural resonance periods of the ocean basins. These waves can be considered modes of &quot;vibration&quot; and can be decomposed using a Fourier decomposition. That is the source of the different tidal constituents that are used currently for tidal prediction.