It sounds like you might not be executing the fights properly. You should be expecting to win all the early fights quickly and decisively, so that each of them is profitable by itself. Hard to say what exactly you're doing wrong, but especially early on the key decision should be what you're targeting, and the key bit of manual execution is timing your shots (do not just use autofire).
At least in early fights you'd want to concentrate on the enemy shields or weapons. A typical way to do that with the starting Kestrel might be e.g. to fire a missile on the enemy shield generator (the missile bypasses the shields, takes the shield room to orange damage, and that takes their shields down completely). Then fire the burst laser at the shield room to take them out for a longer time. After that mainly use the burst laser on their weapon room (to keep from taking any damage), but switch back to targeting the shield room if they get it repaired back to orange.
(As for how quickly you go through, you should leave each level exactly one turn before you're caught by the enemy fleet. Taking longer than that is wasting an opportunity to pick up more resources, taking longer means you're ending up in fights that generate no resources.)