I know some of you may say "it depends", but in the US is a lecturer position inferior to Asst Professor? I am in the job market and get lots of calls for adjunct or lecturer. In the interviews, it is emphasized that "you will be teaching large classes and your job will be just teaching". My question is that if I get into a lecturer position at the start of my career - will that be a bad thing? I do hope to keep applying for tenure track jobs, so will my lecturer experience be considered?
Generally, yes, in the U.S. "lecturer" is a lower-tier, non-tenure-track job, while "assistant professor" is tenure-track. "Inferior" in that sense, yes, although it does happen that assistant professors fail to get tenure and must leave, while lecturers go on and on... admittedly, often term-by-term.
I think that once-upon-a-time any job other than the deluxe ones would have been a sign that you'd already not "made the cut", but nowadays things are so much tighter that it is widely acknowledged that not managing a "research post-doc" or similar is not a disgrace, but as much just bad luck.
It is true that the same factors that push one to a certain threshold so that bad luck becomes the dominant factor will most likely still be in play, so hedging one's bets should take that into account.
Your experience in doing serious teaching will be taken into account in the positive sense that not everyone is able to cope with this, but it is very much the bread-and-butter of academic mathematics, no matter what anyone says. Not glamorous or heroic, no, but...
So when you ask "will my lecturer experience be considered", well, yes, that's legitimate work, and not everyone manages it. But/and if you're applying for somewhat-upscale tenure-track jobs (=assistant prof), those lecturer jobs are approximately neutral. The question will be about what papers you've written/published... given that you apparently pass the minimum threshold of not-awful/intolerable teaching chops.
The most awkward part of such situations is that, often, doing sufficient adjunct/lecturer teaching to make a living leaves insufficient time/energy for scholarship/research/thinking, and this is not usually taken into account formally...
I'm not sure about the U.S., but if your aim is to be a professor that do research, if you accept a position where you are not required to do research (e.g. lecturer), it will certainly not help. To get a professor position where you do research, you need to get some good publication record. If you are a lecturer and do not have enough time to publish and do research, then, yes, it could have a negative impact on your chances to get such position.
By being a lecturer, you would get teaching experience, which is worth somethng but for position where you need to lead a research team, your publication record is generally MUCH more important than teaching experience. In fact, for an assistant professor position, I have often seen universities hiring persons without prior teaching exprience (in Canada).
So in my opinion, it is better to do a post-doc than to be a lecturer if you want to get a professor position where you lead a research team. Being a post-doc will let you continue building your publication record. If you are a lecturer, what might happen is that you may teach too many courses and not have time to publish papers, and thus your publication record may even look worse because you may not publish for a few years, which may leave a huge gap in your CV, and as a result you may never get an assistant professor position.