basic question about CRC and Tcp checksums

This is just a noob question, but I can't find the answer anywhere.

I don't understand how Tcp/IP packets can be insured against corruption with just a 16bit checksum. Hypothetically this should miss 1 in 16000 errors, which isn't much. The reading I have found on the internet says that the ethernet frame has CRC 32bit error checking, which is supposed to mitigate this problem, but my understanding is the ethernet frame is stripped before sending over the internet, which would mean the CRC check doesn't help with any errors accumulated over the transport of the packet over the internet.

Can someone tell me what I am not understanding please?

I need to create a simple messaging program that is guaranteed to be error free, and I am just wondering if I need to create my own checksums for all the data sent over Tcp/IP, or if Tcp/IP can be trusted to be secure?

Thanks.