How do I prove that I arrived at Istanbul airport in time for boarding?

By documenting the evidence at the time it exists. If they don't let you board, insist that they give you something in writing. If they refuse, make a short video that documents their refusal as well as the time and date (there are lots of clocks near the gate).

It's just like when you order something online, you make a screenshot of the order page, because even reputed companies can and will occasionally change order details retroactively, and will only revert the changes if you have proof that they did this.

It takes a bit of paranoia to do this, but that comes very naturally after one or two experiences like the one described by the OP.

Alternatively, you can hope you find a witness. Unfortunately a close friend or family member usually doesn't count, which makes this quite difficult in practice.

Agree with Peter's answer above. Generally you need the following:

• Get the full name and title of the person who refused you boarding (at least to ensure they have the authority to do so).

• Ask for the document explaining your rights for denied boarding compensation, and the paperwork explaining your boarding refusal. Ask to speak to a senior airline representative.

• Talk to a few passengers boarding and if sympathetic, get their contact information, as this would prove you've been at the gate on time.

• Make a video/photos with the boarding gate, boarding line and gate agents. Make several photos and have your passport, visa and boarding pass in those photos to confirm you had the necessary travel documents with you.

• Be at the gate until the boarding ends. If you leave early, the airline may later claim they changed their mind and allowed you to board, but you weren't present and couldn't be found timely (thus no compensation).

• After the boarding ends and you're still refused, take more photos to document this as well.

• Carefully write down the story and all the little details such as who said what. Otherwise you will forget them in a few days.

• Find the airline representative at the airport, and approach them with this evidence and ask for the paperwork explaining your refusal. You'd likely need to go back to check-in for that (but the people you need are not at check-in, so ask for the directions). Do it as soon as possible to make sure they can collect, and document all necessary information. Make it clear you're intend to apply for the denied boarding compensation.

• If you reasonably believe you have been discriminated against because of your race or nationality, mention this too (you may also want to file an official complain with the airport authorities).