What evidence is backing the claim that the CO₂ we're releasing in the atmosphere is the main cause of climate change?
What evidence is backing the claim that the C02 we're releasing in the
  atmosphere is the main cause of climate change?


As you said in your comment:


  I agree that, if the proof is too complicated to be understood by a
  layman, your only choice is to rely on scientific consensus


Complication regarding proof or evidence based consensus in science is tricky.   Should we accept things that we don't understand, that scientists say are so, like the standard model or should we doubt them.   That's a philosophical can of worms that I'd rather not open, but I can say this, if you can't follow the science, perhaps you shouldn't take a side.  Your time would be better spent learning than doubting.   There is no wisdom in naked skepticism.   

That's something many skeptics would be wise to learn.   Skepticism is both great and useless at the same time.   Bart Simpson's a skeptic.   The dumbest kid in the class can say "I'm skeptical about that", and he can answer every question with those 4 words, but he'll never get any smarter.

Applied skepticism is where it's at, and to apply skepticism, you need to understand the argument(s).  Good skepticism requires understanding and, after understanding the argument, then you can intelligently apply skepticism.   Empty doubt is useless.  Good skeptics need to be smart.

This is one argument as to why man made climate science is likely true.   Because there are no good skeptical arguments against it.   Urban heat island - debunked.   Cosmic rays - studied, but no evidence found.   It's the sun?   No, we measure solar output, we'd know.   Mars is getting warmer?  That's incidental.   We're still warming from the last ice age - actually, the opposite is true, we're in a cooling period.  Got any others?   The climate change deniers have figured out that they don't need science to sell their seeds of doubt, they're just selling doubt, but selling doubt isn't a scientific argument.


  What evidence is backing the claim that the C02 is the main cause of
  climate change?


A quick google search will answer this question for you.  Read some of those, and if you still have questions, come back and ask.   This really, really, really has been answered, again and again and again and again.   If you have trouble, I can post some explanations for you.


  CO2 levels have been going up and down for as long as we can look at
  them without human intervention. Eg this NASA graph spanning 400.000
  years


The primary drivers are the milankovich cycle, which leads to the ice-age cycle and feedback mechanisms.  Colder oceans absorb more CO2, warmer oceans release CO2.   CO2 and global temperature have an effect on each other, each driving the other forward or backward depending on how much ice-cover the Earth has.   This is a well understood feedback mechanism that in no way contradicts climate change theory, in fact, they go hand in hand.


  global temperature is still inside the bounds of normal cyclical
  Halocene variations (as seen on wikipedia)


Yes, but the rate of increase is NOT within the bounds of normal Holocene variation.   Also, there is a relation between northern hemisphere summer insolation and temperature.   The direction of the Northern Hemisphere summer insolation is downward right now.  That suggests, without human influence, Earth should be cooling right now, it's not.  It's warming rapidly.

You can say "but the Earth was just as warm 7,000 years ago", and that's about right, but the orbit was different and Earth should have been warmer 7,000 years ago.


  earth temperature has been going up and down for millions of years,
  obviously without human intervention, as seen on the same wikipedia
  page


Hundreds of thousands of years:   Milankovich Cycles

2/5/10/50/100 million years . . . Isthmus of Panama, Himalayas, drifting of Antarctica, Glaciation of Greenland, variation in plant life.

There can be many causes of natural climate change.   One of the ways we know that the current climate change is man made is that we can rule out causes, for example, ocean currents didn't suddenly change in the late 80s.  No new land bridges formed.   Causes are testable and causes have signatures.  CO2 or greenhouse gas driven climate change has a measurable signature.   It's not just that temperature is rising but where it's rising that helps answer these questions, poles vs equator, land vs oceans, upper atmosphere vs lower.


  scientists usually reiterate the first four facts


If you'd read even one scientific study, you'd know this isn't true.  Journalists might mostly reiterate your first four facts, not scientists.


  we know they work because they explain past changes, but this not a
  very strong point since those models have been calibrated using past
  data


There's a principal in science called the double-blind study, where you set up two equal experiments, with one variation and you don't know which experiment has the variation and you study the effects.    This is a good scientific approach but it's not always possible.

You can study the effects of smoking by letting one group smoke and another group not and compare the two groups, but we can't build other Earths to study one with burning fossil fuels and one without - models are all we have.   How accurately the models predict what will happen by 2100 is hard to say, but inaccurate models are better than none.   Take a financial advisory firm that runs models and one that doesn't - and you tell me which one will be right more often.

There's also much more consistency with planet Earth than there is with financial models.  The solar energy the Earth receives year to year varies by less than 0.1%.  Global could cover varies by perhaps 1%-2% year to year.  Snowfall cover varies maybe a little more, volcanic ash comes and goes, but generally leaves the atmosphere pretty quickly, even the largest volcanoes only trigger cooling for a small number of years.   The Earth absorbs heat from sunshine with a small percentage coming from internal heat and volcanoes and it loses heat by radiation.   There are thermodynamic laws that govern heat transfer.    That means the Earth is pretty consistent, having roughly the same orbit around the same star.   By comparison, financial models try to estimate something that's a few orders of magnitude more volatile.  

Modeling isn't easy, but it beats blind guessing like the Patriots beating the Jets.   That's true in both studying climate change and in finance.  

So, how do we know it's man-made?
Look at the signature.

Earth is warmed by the sun during the day and it radiates energy away at night.   The atmosphere acts like a shield, blocking some of the solar energy from reaching the surface and a blanket catching some of the radiation leaving Earth.

Climate change gases like SO2 and very fine volcanic ash thicken the shield, cooling the Earth - see year without summer.   Climate change gases like CO2 thicken the blanket.   Water is also a greenhouse gas so where CO2 has the biggest effect is low absolute humidity regions like the poles.  

Other changes, like albedo, the effect is in visible light, not infra-red.   Ocean current changes are easy to detect because the effects are visible on location.   A stronger mid-atlantic current = warmer European winter.   The climate effects of El Nino, which warms the Earth, are many, but they are well documented.

Heat-trapping greenhouse gas has a signature.   The lower atmosphere warms, the upper atmosphere cools, the tropopause rises and the troposphere loses mass.   The warming can be modeled based on greenhouse gas to be largely polar and the comparing temperature by time of day, the Earth warms most at night, where a hotter sun would do the opposite and warm the Earth during the day and that's where you'd see the larger variation. 

Your question implies that scientists ignore all the other possibilities.   Nothing could be further from the Truth.   Scientists (on average) appreciate possibilities and will rarely dismiss valid alternative explanations.   While the theory was first proposed in the 1900s, modern climate change theory had a rebirth in the 1950s, and was the subject of much study through the 60s, 70s and 80s.   It was met, not with immediate acceptance, but decades of skepticism.   It's only now accepted because the evidence in favor is very strong and the evidence for alternate explanations is hovering close to zero.   Decades of research, models that aren't perfect, but fit both matching what's happened so far and going backward, and the signature of the warming matching what is expected from CO2 and the counter theories all failing.    The evidence is very very strong.

All it would take to disprove manmade climate change theory is one unexplained cold year - just one.  One year like the 1970s without a volcano to cause it and the theory is dead.  That hasn't happened.  We've had nothing but warm years for a few decades now.    You could also disprove it by showing a signature that doesn't fit with CO2.  The opportunity (and funding) to disprove climate change is enormous, but you can't disprove a truth, not scientifically.   If it was false, it wouldn't be that hard to show scientific evidence that disproved it.  It's not like this is Rosalind Franklin who worked out how to take photographs of the DNA molecule and she was reluctant to share them with other scientists who'd take the credit for her work.  This is the Earth.  The data, the evidence is out there.  The means to disprove man made climate change - if it was disprovable, are available, and the funding to disprove it is available and significant and they have nothing.   All the evidence still points towards man made climate change theory being right.

(too long?  too preachy?)