What is the correct method for istinja' to avoid splashing water contaminated with urine onto the body, to avoid excessive handwashing, and to reduce the time spent in the process of istinja'?

It seems you are suffering from waswas (obsessive doubts) regarding purification (taharah), which is a serious matter because it makes worship disliked and burdensome. The best cure for waswas is to ignore it and pay no attention to it, along with frequent supplication and asking Allah for well-being.

As for how to perform istinja' (ablution after relieving oneself) with water: - For urine: Place water on the place where urine exits and wash it gently, reducing the water pressure to avoid splashing. If you cannot reduce the pressure, use a small container or your hand to wash the penis only three times. - For feces: Do the same, being careful to reduce the water pressure. If you cannot, use your hand to wash the impure area. - You can also use tissues first to remove the impurity with three wipes, then use water if you wish. And the splashing of water afterward will not harm you if the area has been purified with tissues.