Hi. I'm Steve. I'm sorry to hear about your TV. I will try my best to give you a precise and accurate answer.
When you say that the tv turns off, does the TV actually go into a standby state or do you just lose the picture?
Most likely the SMPS board.
How handy are you? I'm not sure if replacing it is going to work out cheaper than just purchasing a new 20" TV but it's worth a try. The part number is XXXX likely L2607MPS but there are some minor variations so check the number of the existing board. You should be able to get one for $40 or thereabouts.
I'm sorry there is no quick fix for this issue. I wish there was.
Just ask if you need more help on this issue. I will be here.
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Thanks very much.