The big question is how old is the island.
Rivers are fairly simple really. Any rainfall on your island will flow downhill and collect at the lowest points. Anywhere that enough water collects up while flowing will become a river.
But then you need to add time to the mix. Water flows down to the lowest point, but then it has to evaporate. If it doesn't evaporate as fast as it's flowing in, then the water level rises.
Furthermore, flowing water erodes everything, so a river will eventually carve a channel down to sealevel. ("Eventually" meaning potentially hundreds of millions of years depending on how high the ground started, and how hard the rock is.)
This looks like an island that's at most like a hundred thousand years old that has either low precipitation, or a high evaporation rate.
If precipitation vs evaporation isn't pretty tightly balanced on this island (which will be hard to maintain long-term) then those lakes your rivers drain into will either dry up or fill up. In the former case you now probably have a desert island. In the latter case they either find a way out to the sea, or your island becomes a soup bowl.
That system in the southwest is going to be particularly unstable. According to your coloring the difference in height between the river's close approach to the sea and the inland lake it ultimately appears to drain into is pretty small. One year with extra rain raising the lake level and the river is going to jump its banks at that inlet and carve a new channel. And then that whole section down to the lake is going to reverse direction and the lake will drain into the ocean.
Is it realistic given the contours you've designed? Reasonably. Is it stable? Heck no. The inhabitants of this island are likely in for some major upheaval over the next thousand years or so as the rivers create more stable pathways to sealevel.
Unless it's a volcanic island and there are cracks in the rock under those lakes leading out to the sea, in which case those cracks will erode bigger over time and eventually you'll have sinkholes opening up at unpredictable locations.
None of this is necessarily bad, and may even be preferable for a fantasy story. Plenty of opportunity for shifting borders and rescuing whole villages from sudden shifts in water flow.
But if that's not what you want then think through some fluctuations in water level and what will happen with those collection points. Which way will they go when they overflow? They'll carve a channel to the next lowest point when they do. Repeat until there's nowhere lower for it to escape to.