When I apply the brakes it causes the front wheels to shift and the steering wheel as well.
The condition you have described occurs when one of the car’s front brake rotors is warped. When the high spot on the rotor turns and enters the caliper it slows it down, slowing down that side of the car.
It’s similar to how tanks and tracked vehicles operate: To turn left you slow down the left track and the tank turns left.
Anyway, got a little off topic there. Have your front brakes checked out as it may be time for new pads and rotors.
It sounds as though the front suspension has one or more worn parts.
Applying the brakes causes a shift in suspension parts that is aggravated by one or more of the following: An wheel out-of-balance condition, out-of-alignment condition, or worn parts including tie rod(s), ball joint(s), and/or control arm.
This should be checked by a mechanic you trust and has experience in examining front end suspensions and replacing only the necessary parts and doing only the needed repairs.