This is a common issue with RTL dongles. They have a low dynamic range, ie. they can't receive both weak and strong signals simultaneously, at the same gain level. You need an SDR with a higher dynamic range - Airspy HF+ is said to be much better on this matter, however I don't own one myself, so I haven't done a 1:1 comparison with RTL.
Another, cheaper option is to try a shorter (ie. worse) antenna - given that you're so close to the airport, even that small 10cm stick which often comes bundled with RTL dongles should do reasonably well.