Let's get rid of a red herring from the Wikipedia page linked to in the question!

Horseshoe Lake, Arizona - is a reservoir created by the Horseshoe dam. Nothing to do with actual horseshoe lakes.

The other's lakes listed might be genuine horseshoe lakes, bu there are no aerial photographs of them.

Generally, horseshoe lakes are oxbow lakes which is

a U-shaped lake that forms when a wide meander of a river is cut off, creating a free-standing body of water. In south Texas, oxbows left by the Rio Grande are called resacas. In Australia, oxbow lakes are called billabongs.

A resacas is a

former channels of the Rio Grande and are naturally cut off from the river, having no inlet or outlet

The Rio Grande's water moves from Colorado to the Gulf of Mexico. Every year during spring, melted snow would flow into the Rio Grande, bringing seasonal flood waters to the most southern tip of Texas.3 Given the overflow of the river's main and distributary channel banks, the Rio Grande would carve new river channels, resacas. When the seasonal flooding ended and the river retreated, the newly formed oxbow lakes remained, creating resacas all throughout the Rio Grande Valley.

In Australia, billaongs are

an isolated pond left behind after a river changes course.1 Billabongs are usually formed when the path of a creek or river changes, leaving the former branch with a dead end. As a result of the arid Australian climate in which these "dead rivers" are often found, billabongs fill with water seasonally but can be dry for a greater part of the year.

Billabongs were significant because they held water longer than parts of rivers; it was important for people to identify and name these areas.

In general,

An oxbow lake forms when a river creates a meander, due to the river's eroding bank. After a long period of time, the meander becomes very curved, and eventually the neck of the meander becomes narrower and the river cuts through the neck, cutting off the meander and forming an oxbow lake.

When a river reaches a low-lying plain, often in its final course to the sea or a lake, it meanders widely. In the vicinity of a river bend, deposition occurs on the convex bank (the bank with the smaller radius). In contrast, both lateral erosion and undercutting occur on the cut bank or concave bank (the bank with the greater radius). Continuous deposition on the convex bank and erosion of the concave bank of a meandering river cause the formation of a very pronounced meander with two concave banks getting closer. The narrow neck of land between the two neighboring concave banks is finally cut through, either by lateral erosion of the two concave banks or by the strong currents of a flood. When this happens a new, straighter river channel develops—and an abandoned meander loop, called a cutoff, forms. When deposition finally seals off the cutoff from the river channel, an oxbow lake forms. This process can occur over a time from a few years to several decades, and may sometimes become essentially static.

The naming of oxbow lakes as horseshoe lakes makes just American terminology. When such lakes were being "discovered" and "chartered" most people would have known about horseshoes, but maybe not oxbows so they got named after what they knew.

Oxbow lakes do exist elsewhere, they're just not named as horseshoe lakes.

The Kanwar Lake Bird Sanctuary in India is one of the largest oxbow lakes in Asia. In the United Kingdom they also occur in the Gower Peninsular of southwest Wales and there is also the Bole and Burton Round in Nottinghamshire.

Celestially, there has also been a possible oxbow lake postulated in Saraswati Flumen near Ontario Lacus on Saturn's moon Titan