My website got hit by a Google update last year around May time, and then another in July. The site went from around 10,000 daily visits to around 8,000, then to around between 4,000-5,000, which has been maintained since then.
The site had a massive rise between 2011 - 2013, where we went from having between 1,000 and 2,000 visits per day to around 10,000 (the visits increased in increments, not all at once). We were getting 10,000 a day for around 6 months before the first update hit. All SEO was totally white hat - the site and content has been worked on for many years (the site is almost 15 years old) and it finally paid off over that 18 month period.
All of our pages still seem to be there in the Google listings, but they have just moved down a few places. For instance, for our main keyword (which is very competitive) we have moved from spots one to 3 (we used to fluctuate), to the bottom of the first page or sometimes the top spot of the second page in Google. There is a similar pattern for many of our keywords.
We have not had any messages from Google in Webmaster tools with details of a penalty.
I have been in discussions with an SEO agency and they say that the first step (that has worked for many of their clients) is to 'cleanse' our backlinks. They say that they will try to get rid of all of the bad or useless ones (by contacting the websites) and they also say that our 'branded' keywords vs. anchor text keywords are not balanced (there are an unnatural amount for our main keyword vs. our brand online) so they will try to tidy this up and change some of the keywords to brand words so that the balance becomes better. They say that a cleanse is very important.
Please note we are also working on a big website update and improving things across the board, which of course is the most important thing, but do you think that what the SEO company have advised could help? I was always under the impression that bad links don't have any negative impact - if they did competitors could build lots of bad links against your site to ruin your rankings!
Thanks in advance!