Problem Powering 13.3 inch LCD Screen with 3.7v Battery
The screen spec says "Input Power Adapter: 5Vdc-12Vdc More than 2A" - which I presume means that it may draw up to 2A. The XL6009 boost converter is rated for a maximum switching current of 4A. The 4000mAh Lipo cell has no current spec, but it can probably do 0.5C (2A), and the 2mm JST PH connector on it is rated for 2A.

So it should all be good, right? Not quite. 

Watts = Volts x Amps. Your screen needs up to 2A at 12V, which is 24W. 24W/3.7V = 6.5A. In practice the booster might only be 80% efficient, so the battery current might need to be 25% higher again, ie. ~8A. That's way more than the battery and booster can handle, so your setup does not come close to meeting the specified power input for your screen.

However you found that it actually only draws 0.6A after starting up. So lets do the calculations again. 12V x 0.6A = 7.2W. 7.2W/80% = 9W input power. 9W/3.7V = 2.4A. The booster should be able to handle 2.4A, but the battery connector (and wiring?) won't. So even if the screen never drew more than 0.6A your battery would still be overloaded.

You need a better battery, and perhaps a more powerful booster. One solution might be to get an extra cell and wire it in series to make make a 2S (7.4V) battery. This would halve the battery current, and might be enough to run your screen properly if it never drew more than 1A. The 2S battery would also last twice as long per charge, but would need an appropriate charger.