So when I start my 2003 Discovery in the morning and put it in D, and it shows the letter "D" in the dash. The first 5km it shifts and goes just like should, but after that the "D" disappears from the dashboard and it starts shifting lazily and sort of feels like the transmission is slipping?

On even slight uphills it often doesn't want to shift and accelerates very poorly no matter how high the revs are.

When I turn it off for 30min-1h, and go on the road again it might go for 1km without any problems and then the problems start again.

The engine itself show no symptoms, only the transmission really.

This may be the wastegate. Ensure that it is properly attached and that the nuts are done up tight.
This problem will reset every time you turn the ignition off but reappear every time you try to accelerate, or drive up a hill. It is because if the wastegate mechanical connection is not right the ECU reads it as an overfuelling situation and cuts back on fuel, putting it into "limp home" mode to protect the engine. But it does seem as if it is a gearbox issue.
BUT It may be something completely different. But look at this fiorst and get back to us.
Nice to meet you by the way, welcome to the forum and when you have a minute why not go to the "Introduce yourself " section???!!!
Good luck.

Gearbox switch or loom?

What's the oil condition and level like?

If the D dissapears the problem is with the XYZ switch's output but that should trigger the flashing M+S lights... can you see those warnings on ignition self test?

Sorry about abandoning this thread, completely forgot about posting it. After almost 3 years I'm back though

The problem was the XYZ switch, oxidation had gotten into the contact surfaces. This fix wasn't as bad as I expected, taking the switch out was the hardest part as it is in a tight spot behind the exhaust pipe and you can't really see what you are doing with your hands.

I drilled one end of the aluminium pins that hold the casing together so I could take it apart.

I cleaned the contacts inside. 2 of the contact "fingers" were bent out of shape slightly so they weren't holding proper contact, I carefully bent those back.

When I started to put it back together, I sprayed the insides with some electronic contact oil. I replaced the aluminum pins with small screws and nyloc nuts.

I put the switch back in the car no problem, had to readjust its position slightly after short drive though.
Here are a couple pictures from August 2020, I don't have pictures of the finished product, sorry.

I have a little trick to deal with failing sector switches, limped one in a previous car well over a year doing this.

It played up when cold and reversing.

Always reverse in and drive out.

So warm less likely errors.
If it errors you are turning it off anyway.