Why are there some proteins that have a known amino acid sequence, but their 3D structure is not known? Wouldn't finding the former in a lab lead to the discovery of the latter? Please correct me if I have misunderstood something.
Protein sequencing is a nicely constrained problem: you have a one-dimensional sequence of amino acid members, which come from a limited set of options (made a bit more complicated by post-translational modifications, but not much more so). Because it's one-dimensional, it's a problem you can easily solve by chopping up a protein into little bits, using mass differences between amino acids to understand their constituents, and determining the order from that distribution. If a DNA (or mRNA) sequence is known, it becomes even easier - you can skip the protein sequencing process and get the amino acid sequence directly from the nucleic acid sequence and the genetic code.
By comparison, protein folding is an absolute nightmare to solve for. Chemical bonds between amino acids are not rigid, they can bend and twist in all directions. The conformation of those bonds also depends not just on adjacent amino acids (as in a 1-D problem) but potentially on any other amino acid in the sequence (not to mention external influences..).
In a large molecule like a protein there is a massive massive degrees of freedom problem. From Wikipedia, describing Levinthal's paradox, bold mine:
In 1969, Cyrus Levinthal noted that, because of the very large number of degrees of freedom in an unfolded polypeptide chain, the molecule has an astronomical number of possible conformations. An estimate of 3300 or 10143 was made in one of his papers (often incorrectly cited as the 1968 paper). For example, a polypeptide of 100 residues will have 99 peptide bonds, and therefore 198 different phi and psi bond angles. If each of these bond angles can be in one of three stable conformations, the protein may misfold into a maximum of 3198 different conformations (including any possible folding redundancy). Therefore, if a protein were to attain its correctly folded configuration by sequentially sampling all the possible conformations, it would require a time longer than the age of the universe to arrive at its correct native conformation.
Now, of course that's not the actual process that proteins use to fold (they don't iterate through all possible combinations, they settle through an energy landscape where only certain intermediate conformations are realized), and we can use that in computational models to solve protein structures more quickly than the age of the universe, but it's still quite a slow process. Projects like Folding@home have aimed to distribute the computational load among unused processing power in devices around the world, including idle gaming consoles and personal computers, but there are many many protein structures to solve.
It's possible to get a general picture of protein shape using imaging techniques like X-ray crystallography or cryo-EM, and for some purposes these techniques give a lot of information, but these techniques are also by no means simple and can be prone to errors.
To answer why sequences are known before structures, it is worth highlighting the typical ‘workflow’ for a biochemical researcher. Briefly, sequence is always before structure because you need the sequence to determine the structure. As with everything else one would like to investigate, you have to start with the information that you already have. In modern sciences these are usually as following for protein research:
1. Isolate some bacteria or fungi from e.g. the ocean or anywhere else, and sequence their whole genome (DNA). This is very realistic to do, and not that expensive anymore.
2. Once you have the genome sequence, there is a lot of bioinformatic work to be done to annotate the sequence. This means, identify coding regions for e.g. for proteins. There are programs that are very good at this, because we already have a lot of information on what is encoded in living organisms (based on experimental data and years of research).
3. The DNA annotation programs can assign thousands of proteins in one genome. These protein sequences are then uploaded in relevant databases, for other people to view and work with. Note that these proteins sequences are NOT experimentally validated. They are however assumed to be correct with some statistical validity (which is usually correct now a days due to the overwhelming amounts of collected knowledge, and sophisticated software).
4. Scientists (i.e. biochemist and others) can then work with these protein sequences to find out if they actually are what the programs assumes. This involves the bottleneck of actually been able to produce and purify the protein of interest (which may be very difficult).
5. After experimentally assuring that the protein actually has the function you are interested in (by doing experiments), and being able to actually produce and purify it, one would typically want to determine its structure. This is because the three-dimensional structure of a protein can explain how and why it work the way it does. This is however difficult to do, experimentally (as well described by the other post answer).
Can you trust homology-models?
As a scientist working with protein structure and function, I would also note that (in my opinion) you cannot completely trust structures solved purely computationally (i.e. homology-models). These estimated homology-models are simply based on actually structures that are experimentally validated (e.g. crystallographic). Even though homology-models are very useful when you do not have a better structure, you can never be completely sure that they are correct (as they are simply assumed models of structural models; i.e. models of a models).
The active site of enzymes is of great interest to understand how enzymes catalytic their reaction. It is of vital importance to know how the catalytic residues are structurally placed/arranged in the active site to understand and even modify their catalytic behaviour. Even if you have a homology-model is 98% correct, the 2% error could be the structural placement of catalytically important residues. You also can not know for sure what the error is. One should therefore be very careful to put to much reasoning into a homology-model. In summary, if you don’t have a experimentally validated structure (which is difficult to get) you can never be completely sure of what is going on (or at least, you would be working in the dark - looking for effects).
Experimentally validated structures:
I would also like to add that x-ray crystal structures are, as of today, the golden standard when it comes to protein structure information (although cryoEM is catching up(!), and NMR structure give a lot of information about dynamics). You should check out the PDB database. If you have a high resolution structure, of e.g. 1.1A you are approaching atomic resolution and can even see the rings in the aromatic amino acid side chains (which is very cool!).
To answer your question in brief:
Sequence is always before stucture as you cannot experimentally determine structure without the sequence (also required to computationally model the structure). The protein sequences are simply assigned with complexed programs, from the DNA sequences. After you have the sequence you need to experimentally validate that the assigned protein sequences are correct. Only after all this work ca you start to determine its three-dimensional structure.....through a lot of hard work.