Redrobes' info is great, but if you're not getting it from the tutorial, let me try some different words. First off, is maybe a component of your troubles an inability to imagine 3-d views in your mind's eye? If you can, that makes imagining which way is downhill easier. If not, you'll have to use more rules of thumb and generalizations.

Some of those rules:

Figure almost every stream is going to wind up as a river entering the sea. Imagine some of those little short riverlets you show as looking around for "a way further down". If there truly is no lower land nearby, well that water and all his friends are going to hang around and party - call that a lake - until the crowd gets so big that the outer edge gets above the rim of the local lowland, and starts to flow again.

Yup, there are streams that disappear, either by ducking underground and continuing to flow, or by fizzling out into an arid wasteland. Those are pretty rare special cases, so for believability's sake avoid them unless a specific story line calls for them.

Another good general rule- rivers don't split. They join. At the scale of continental map you have, it'll be a rare case for there to even be a brief divergence visible that you'd call an island; any such multipath watercourses - termed threaded streams- are going to be within the width of the line you use to show a river. You definitely don't see such splits as your westernmost river, where you show two widely divergent ways to the sea. Using that particular river I can illustrate another placement generality -- that inlet is there because it's lower than the two peninsulas that you have water running along the length of. Figure the peninsulas must be ridges, at least generally. So you would expect water to "fall off to the side" of both of them instead of running the long way down the middle. Which is incidentally why someone asked you about scale - if those are about the size of the whole Spain + Portugal peninsula, okay, I'll buy there being enough room for slightly higher land on the north and south edges, "channeling" the water out toward the tip. If the scale is more like those are ten miles across or even a hundred, my instincts are to not buy it. Viewers don't do all this analysis, they just instinctively feel something is right or isn't. Some of course being clueless, some having a vague discomfort, and some experiencing the same sensation as fingernails on a chalkboard. I'm in the latter camp. That's because I do have a bit of a 3D picture even just based on your coastline, so those two river courses west of the split look to me like they're literally running along ridgetops.

The smaller one just south of that major divergence also shouldn't split, but I'll use it to illustrate another generality: water insisting on flowing downhill means you'll never see it flowing *across* a hill. Now if your scale is small enough ("smaller scale equals bigger area shown" is a mnemonic) there could be a valley across the foothills of that range. But the generality would be for us to expect the overall slope at any point to be away from the pictured mountains straight toward the nearest coast. So if you need a river to skirt that range just so, you want to indicate foothills or something to push water back toward the steep mountains.