How can a child be disciplined without hitting or shouting?

With clear rules, reasonable and consistent consequences, and example — hitting is not the main method in the sunnah.
The Prophet ﷺ never struck anyone with his hand, neither a woman nor a servant, except in battle in the path of Allah (Sahih Muslim, Kitab al-Fadha'il), and Anas r.a., who grew up in his care, was never rebuked even with the word "uff" (Sahih al-Bukhari 6038). In practice: set a few rules the child understands; explain the reasons; give consequences related to the misdeed, for example the device is taken away if screen time is broken; praise good behaviour specifically; and make sure both parents agree. Do not strike the face, hit in anger or shame a child in front of others. If you often lose your patience, that is a sign you yourself need rest or support. See Educating the family and the question on the cane and prayer.