Organic (in the sense of natural) insecticides are sometimes good but they are never as effective as available synthetic solutions. In the few studies done about this issue, (mainly dealing with cabbage) all of them have found synthetic solutions much better at controlling insect pests. However, garlic based insecticides have proven effective at controlling insects if not completely removing them from the garden (they also prove almost non fatal to beneficial insect populations).
In order to control your insect populations easily, you can manufacture an insecticide with garlic, vegetable oil and water. Cut 3 cloves of garlic into small pieces and place them in a jar with 200 mL (about a cup) of vegetable oil for two nights (this ensures that the organic non polar molecules present inside garlic are extracted efficiently). Next, mix this with 2/3 of a gallon of water . Agitate strongly before each use. Then your insecticide is ready to be spread on your garden.
Hopefully, if spread adequately over every inch of the plant (including the underside of leaves), the insecticide should prove efficient against most types of bad insects. It also proves effective in being resistant to rain so your plants should remain protected for a prolonged amount of time. Bear in mind that soap or detergents should not be added to improve surface tension properties because this increases the capacity of rain to remove the insecticide effectively.