It is the wisdom here that rivers do not split (or so I recall it being the basic advice to map-makers). I wanted to give a few local (to me) examples of rivers that do split, and the geological explanation as far as I understand it.

In Ontario, Canada, the South River flows into Lake Nipissing, from there emptying into Lake Huron and after a few more Great Lakes the St Lawrence. The Amable du Fond shares a significant part of its watershed with the South River but it splits and then finds a more direct route to the St Lawrence by eventually emptying into the Ottawa River, skipping all the Great Lakes.

Similarly, the Magnetawan River experiences a kind of fracturing, where it splits into the Magnetawan and the South Magnetawan. The South Magnetawan partially flows back into the Magnetawan, but another channel of it flows into the Naiscoot River. The Magnetawan and the Naiscoot then both flow into Lake Huron.

Two geological factors conspire together to produce this outcome. Both of these splitting rivers lie on the Canadian Shield. The Canadian Shield is a name for the large layer of granite that is very close to the surface under much of Canada's north. This leads to a visually distinctive landscape, but also a certain resistance of the landscape to erosion. Canada has a large proportion of the entire world's liquid freshwater in part because the water collects on the pockmarked landscape of the Canadian shield (this is quite visible on topographic maps if you compare Ontario north of Lake Simcoe to the southern parts of the province), in this landscape it takes a long time for water to create fast-draining water-eroded valleys. This resistance to erosion means that a lake with two outlets will not necessarily begin to favour one over the other quickly, because neither is able to erode a deeper channel compared to the other particularly quickly (this is particularly true for the rivers in my examples, which for the most part are relatively small). This factor explains how a split, once it exists, might remain stable for awhile. The other geological factor contributing to the existence of splitting is that quite recently the entire region was under glaciers during the last ice age. In that time, there were not watersheds in the same way that there are now. So not only does it take awhile for one river to cut off its splitting competitor, but they haven't had very long to try. These two factors combined make for a decent number of splitting rivers.

You can see the ways in which the river is forced to follow the landscape, rather than the landscape following the river, by looking at the Magnetawan River and similar rivers on the Canadian shield like the French River on topographical maps. They have a distinctive jagged appearance that really differs from other rivers, with long sections where they split into several different channels before rejoining (or not).