Why is the steering so twitchy? Steel 90's MTB fork
Short answer - your fork is stuffed.  Stop riding it.

Longer answer - Here's your bike with the main lines drawn on top



And compare that with this approximate equivalent Peugeot bike.  Spot the differences and compare with my list below...




Headtube angle is steeper on your bike - closer to vertical
Negative trail -


A line drawn straight down from your front hub hits the ground at a point.  This is where the tyre touches the ground.
A line drawn down your head tube center line hits the ground after the first line.
This difference is called trail, because the tyre's contact patch "trails" behind the head tube's axis.  Normally, the larger the trail, the more stable your ride is and the more effort it takes to turn. The trail is what helps a bike go straight by default.
Your bike doesn't have positive trail... it has negative trail, or "lead" if you like.  Your bike's head tube angle is continuously pushing the front wheel to a side and you have to actively correct this.  

Additionally, notice the top tube slopes down.  That was not done on MTBs where the top tube was horizontal or sloped up to the head tube.  Since the fork is stepped forward at the end, the front is now lower.  As a compromise, someone has fitted a small A bar from a kid's bike into the quill stem.  This weird set of angles will also compromise handling - you've effectively got 20+ centimetres of stem and the handlebars are noticeably higher than your saddle.




Upshot - you're riding a clown bike that is trying to murder you and may fail at any moment with little to no warning.



The fork is your problem - it needs replacing outright.  Once that's done, the existing wheel and brakes should fit up normally.  I'd strongly suggest getting a flat bar not that double-stack - once the fork's fixed then the extra height will be more of a hinderance and you won't want to be sitting up like a dutch bike with your arms straight forward.

Do check the headset bearings too - there's a chance they've been damaged.

There appears to be too much bend there for a safe un-bending.  Steel is nicely flexible but it has bend limits, and you've likely exceeded them with that level of distortion.   If you continue to ride this, the fork is being extra-stressed in the back under the headtube, and this is where it will fail first.  

Please ride safe bikes.  Right now, this is not a safe bike.   This bike can be safe again, with work.