For how long should I cook onion paste?
Pureeing onions makes them release a lot more of their water than just chopping them, and if you added extra water on top of that, you probably had a very wet paste. That makes it take a lot longer to cook - while you're just boiling away water, it's not hot enough to brown much.

So cook it as long as it takes - it's not going to become bitter. (If anything, it'll get sweeter!)  And if you want it to cook faster, don't add extra water. If you watch carefully, you can also turn the stove up hotter at the beginning to boil off water faster, then reduce the heat once it's drying out enough to avoid burning it.

If whatever you used to make the puree doesn't work well without the extra water, you can still use it; just know that it'll make it take longer to cook. Otherwise, you could try grating the whole onion on a cheese grater, which does a surprisingly good job - it'll still be really juicy though!