There has been much talk that there must be corruption and fraud in the electoral system - and thus that the election of the current Congress was illegitimate because a majority of Americans disapprove of this Congress' actions. It is certainly possible that this is the case; however, that conclusion cannot be drawn simply from the fact that a majority of Americans disapprove of this Congress; instead, additional evidence of corruption and fraud must be shown before one can rationally conclude that the last elections were illegitimate because caused by corruption and fraud.
To illustrate that this can happen in a legitimate election I'm going to make some simplifying assumptions that are fully consistent with reality:
1) assume that Congress consists of one house, with a total of 100 representatives.
2) assume that there are 100 congressional districts, each of which elects one representative to Congress, and that each district contains the same number of voters.
3) assume that there are 2 political parties - R and D - and that every voter belongs to one or the other.
4) assume that all of the voters in each district vote the same way - that is, each representative received 100% of the votes in his/her district.
5) assume that Congress acts when 50% plus 1 of the representatives vote to authorize a particular act (generally speaking, that means that all laws are passed by simply majority).
6) assume that an individual disapproves of Congress as a whole when that individual disagrees with more than 50% of the actions taken by Congress, and further, that an individual disagrees with an act of Congress when that act is approved of by the other party.
7) assume that every individual voter always approves of his/her representative; further assume that an individual approves of her/his representative if that representative votes with his/her party more than 50% of the time.
8) assume that as currently constituted Congress is evenly divided between the R party and the D party - i.e., each party has 50 representatives.
9) assume that each representative always votes Yes or No; i.e., there are no "present" votes and no representative is absent from any vote.
10) finally, assume that each Congressional representative usually, but not always, votes with his/her party; i.e., the possibility that a representative may dissent from his/her party exists.
Under these conditions, Congress can only act if a vote of the representatives is at least either R-51 & D-49, or R-49 & D-51. That implies that each time Congress acts, there was at least one dissenter who voted with the other party.
Further, since the representatives, and therefore the Congressional districts, are evenly split between the two parties, every time Congress acts, at least 51% of the electorate will disapprove of that action. To see this, consider a vote where one D dissented from her party and voted with the R party, giving a result of R-51 & D-49. Now, since the R party won, all voters in the D party will disapprove of that act; therefore, 50% of the voters disapprove of this act of Congress.
However, because only one D dissented from her party and voted with the R party, the voters of only one district disapprove of their representative on this particular act of Congress; the voters in the other 99 districts approve of their representative's vote. That means that 99% of the voters approve of their own representative, even though 50% of the voters disapprove of Congress on this particular act of Congress.
If we assume that (a) this Congress ends up approving 100 acts, each by a 51/49 vote, 50 of which were D party acts and 50 of which were R party acts, and (b) each representative dissented once, and only once, from his/her party, then at the end of this Congressional session we have a situation where 50% of the voters have disapproved of each vote even though 99% of the voters have approved of their representative on each vote.
Nonetheless, in this situation we end up with a case where the voters still approve of Congress - but only by a slim margin - because disapproval requires that the voters disapprove more than 50% of the time; we can, however, already see that there can be a radical split between how many voters approve of Congress as a whole and how many voters approve of their individual representatives.
Now, let's change the assumptions very slightly; namely, assume (1) that the D party controls 51% of the Congressional districts and the R party 49% of the Congressional districts, and (2) in every Congressional vote two D party representatives dissent, that each individual D party representative never dissents on more than 2 votes, and that no R party representative ever dissents.
Under those circumstances, every Congressional action will be authorized by a vote of 49 Rs and 2 Ds, which means that every act of Congress will be an R party act. In turn, that means that the voters in 51% of the Congressional districts will always disapprove of every act of Congress. However, because only 2 D party representatives dissented on each vote, that means that on each vote the voters in 98 of the Congressional districts approved of their representative and, accordingly, that 98% of the voters approved of their own representative.
Finally - and in particular - each and every D party voter has a 98% (or higher) approval rating of his/her representative because that representative only voted with the R party at most 2 times. Because no R party representative ever dissented, every R party voter has a 100% approval rating of his/her representative.
Consequently, we now have a situation where every voter overwhelmingly approves of his or her representative, but the voters as a whole disapprove of Congress because 51% of the voters always disapproved of each and every act of Congress, even though each and every act of Congress was approved on a simply majority vote.
What I want to emphasize is this: we get this result even though - by your own premises - there was no corruption or fraud in the election system because each and every representative voted in accordance with the wishes of his/her constituents at least 98% of the time (each D at least 98% of the time and each R 100% of the time).
So, without any corruption or electoral fraud - each voter overwhelmingly approves of his/her own representative - we get a situation that mirrors what you're complaining of: a majority of the voters disapprove of Congress. However, we also get the situation that I described: majority disapproval of Congress coupled with overwhelming support for each and every member of Congress.