People must understand that the US political spectrum is considerably “right-shifted” relative to the global political spectrum. Thus, it does not surprise me that many in the US who are even just mildly center-right (in the US context) would view the BBC as being left-of-center. I can assure you that from a global perspective, they are not; they are very scrupulously centrist.
Even just within the US context, people who are pretty far out on one wing or the other seem to see bias where it might not be. I have heard people on the right complain that the NY Times is a leftist rag, while I have heard people on the left complain that the NY Times is hopelessly conservative. Obviously it cannot be both, and honestly it is neither — maybe very slightly center-left on the US spectrum, but very much centrist globally.
I would suggest three things:
First, get your news from a variety of sources, both domestic and foreign, and across the political spectrum, but concentrated in the center (both US and global).
Second, realize that much immediate news reporting will end up being wrong, and in any case breaking news tends to crowd out important long-term news. You really need to make a deliberate effort to balance your consumption of daily news with weekly and monthly periodicals (or their on-line equivalents), especially of the more in-depth and analytical type. A steady diet of reading of current affairs books is also useful.
Third, you must develop the ability and habit of thinking independently and critically. It greatly helps to know a lot about history, political science, sociology, psychology, geography, economics, science & technology, etc. People who are very knowledgeable about these things are capable of independently evaluating what they hear or read in the news, and are quick to spot the falsehoods and distortions. If talking heads are getting these old things wrong, maybe they should not be trusted to have the new stuff right either.