What you say is not always correct, there are several circumstances that can cause rivers to become less wide, or even disappear entirely.
If the riverbed is made from hard materials, then the width of the river is mostly decided by the width of the riverbed, rather than the amount of water flowing in the river.
In the places where the river is thin the water then has to flow faster, and it does. I believe that is where the term 'rapids' come from, the water flow much faster there.
In areas where the ground is dry rivers often carry less and less water as they flow. Partly because water evaporate from the river, and partly because water sink into the ground raising the water table a bit.
But mostly you are correct. the majority of rivers do get wider as they flow towards the sear. the reason for this is that rivers generally come from rain or other precipitation, and that falls over a very large area. that water falls and after awhile if forms puddles and little streams flowing towards lover areas.
And because all the streams flow toward lover areas they tend to meet each other so they form a bigger stream, eventually so many little streams have joined that we call it a river.
And it does not stop there, streams and even other rivers keep joining our river filling it up with more and more water, and making it wider and wider. such a smaller stream or river joining a big river are called 'tributary' because it contribute in making the big river even bigger.
North Sea and Mediterranean Sea
It gets wider all the way to the sea, like most rivers.
North towards the Arctic sea
this part of a river is called the mouth of a river because it opens up into the sea and while it is doing that it is groing wider and wider and wider. sometimes it douesn't have to be running into the sea it could be a lake that then joins onto another river or another wide or large water area like that.
Some rivers have a lacustrine source but most river heads rise a springs. Rivers flow towards ocean sea level, they do not arise there.
All rivers flow down slope and usually towards the Sea.
Because the oceans are at sea level and the rivers are always above sea level, meaning that water flows downhill.
Streams and rivers are the result of rain falling to the ground. The water runs downhill and flows towards the sea.
The early explorers believed that a great inland sea lay in the centre of Australia. They believed this because the rivers did not behave as rivers in other countries did. Australia's rivers flowed away from the coast, not towards it.
Australia's rivers are unusual because they flow away from the coast rather than towards the coast. On all other continents, the rivers flow towards the sea. In Australia, a great number of them flow inland, either towards the Murray River, which eventually empties into the Southern Ocean, or into the inland salt lakes.Many of the inland rivers flood during the wet season, yet can dry up later in the year. The coastal rivers do not run dry.
Sea fish don't live in rivers because rivers don't have coral or any sea plants. The sea plants are vital for some fish. That and some of the sea fish are to big for rivers.
Many of the rivers of eastern Australia flow away fromthe coast, rather than towards the coast. They do not empty into the sea like rivers in most of the rest of the world.