Stop led from blinking when running
I tried your sketch on an Arduino Uno, with no servos connected, and the LED on pin 13 did not blink.

DataFiddler makes a good point in their comment. To be sure you are actually uploading the sketch, try uploading the Blink sketch that comes with the IDE. Change the delays from 1000ms to 200ms then upload it. The difference in flash rates should confirm or deny the ability to successful upload sketches.

You don't say which servos you are using, so I'll assume they are SG90. I'm not sure if the data sheet for every SG90 servo is the same, but if you look at this one, SG90 Specifications, you'll see the "stall current" can be between 570mA and 730mA, each. If you are powering them from the Uno, it can't supply that much current. Another limiting factor is the computer's USB maximum current output, (if you are powering the Uno with a USB port). FWIW, my computer can only provide 500mA per USB.

I would suggest powering the servos with a separate power supply, then see what happens.