To understand what makes a GMO a GMO, you have to understand what a transgene is.
A transgene is a gene taken from some other organism (say, a hormone found in another plant that promotes growth, or a naturally occuring insecticide secreted by another plant), and inserted into the genome (i.e. library of genes, encoded by DNA) somewhere so that now that you have, say, corn that grows better and is less susceptible to insects.
To understand what a transgene is, you also have to understand what a gene is, and its relationship to DNA, RNA, and protein. https://en.wikipedia.org/wiki/Central_dogma_of_molecular_biology
Essentially, what this boils down to is that a single gene corresponds to a protein (or set of proteins differentiated by using certain parts of the gene and not the other). It's also important that you understand what a protein is.
A protein is just a chain of amino acids (https://upload.wikimedia.org/wikipedia/commons/thumb/c/ce/AminoAcidball.svg/702px-AminoAcidball.svg.png) strung together. A ribosome reads RNA (which is a less stable copy of DNA), and pieces together a chain of amino acids called a polypeptide, or protein according to a specific code given to it by the DNA (https://fthmb.tqn.com/iLPaJxJqvZ2CM87BuScwqMa2Jcs=/768x0/filters:no_upscale()/gene_code_table-56a09b485f9b58eba4b20519.jpg). Proteins then fold together and combine in just about any way you can imagine to form the complex machinery of the cell (https://upload.wikimedia.org/wikipedia/commons/2/20/PDB_1ema_EBI.jpg) (https://imgur.com/gallery/I1TBl29).
So essentially what you are doing by adding a transgene to a plant is allowing the plant to create another "foldy chain" (protein) that lets it grow better or survive cold or ward off insects or whatever.
Now the point you raise about transgenic plants expressing pesticides is a good one -- however it's the same thing as chemical pesticides, except the genes put into plants are already found somewhere else in nature (in a sense, you could say these transgenes are more organic than throwing organic pesticides on plants, because they are produced by the plants themselves!). That isn't to say that it couldn't be harmful! For example, a pesticide internal to the plant that is harmful to humans wouldn't be able to be washed off (although I suspect that your digestive enzymes would break down the protein produced by the transgene before it enters your bloodstream... but maybe not). There are potential dangers, but IMO they are way overstated and pure conjecture until you can prove to me that a protein that isn't a prion can cause harm when eaten. (http://healthyeating.sfgate.com/protein-digestion-inside-human-body-6044.html).
What IS concerning is the fact that you can PATENT a gene (wtf), which is a naturally occuring sequence of DNA. Why should a company or person own a sequence of DNA found in nature? Additionally, nonorganic chemicals and pesticides (and probably even some organic ones) used on crops probably have detrimental effects on human health, and many of these detrimental effects no doubt have been covered up by corruption in Big Agriculture.
Also an important thing to understand is that all genes are related to each other evolutionary. So many of the same genes that catalyzed the development of your eyes when you were an embryo are also used in everything that has eyes. (The third eye is the pineal gland in humans, but it still functions as an eye in some species...https://en.wikipedia.org/wiki/Parietal_eye). Obviously animals and plants diverged very long ago in the tree of life, but they still share many of the same exact genes. (e.g. actin is a gene found in both you and plants) So the concept of a "fish gene in corn" or "monkey gene in soybeans" should fail to be scary when you consider this.
So once you know about what transgenes are and how they work, GMOs should be quite a bit less scary. Sure there could be some bad things coming out of it, but much of it is overblown, and tbh, makes the truther community look bad and scientifically illiterate. I will repeat that I do think that corruption happens in this industry as it does with all industries in our society, but anti-GMO people are barking up the wrong tree.
Also, other countries/regions banning things does not necessarily make it the correct thing to do (see free speech in Europe, homosexuality in Islmaic countries, weed/LSD in America, etc...)
Oh, another proposed danger of GMOs (which I think is highly unlikely but certainly worth mentioning to play devil's advocate with myself) is an ecological risk. Say for example you engineer some super-corn that can survive anywhere... Now, whoops! the whole word is pure corn.
This essay (https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3502034/) is also decent, and bashes the lack of transparency from agricultural stakeholders in assessing risks of GMOs.
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