The Panda algorithm update named "Top Heavy" first issued in January 2012, then October 9 2012, and again February 6 2014 with various unannounced updates along the way specifically targets sites that are "top heavy". While the discussion is primarily focused on sites/pages top heavy with ads, Google states in the January 19 2012 announcement of the Top Heavy algorithm:
As we've mentioned previously, we've heard complaints from users that
if they click on a result and it's difficult to find the actual
content, they aren't happy with the experience. Rather than scrolling
down the page past a slew of ads, users want to see content right
away. So sites that don’t have much content “above-the-fold” can be
affected by this change. If you click on a website and the part of the
website you see first either doesn't have a lot of visible content
above-the-fold or dedicates a large fraction of the site’s initial
screen real estate to ads, that’s not a very good user experience.
Such sites may not rank as highly going forward.
How far Top Heavy goes for sites that wide header space but is not top heavy with ads, is anyone's guess. However, be warned that Google is updating it's rendering engine for just this purpose. The intent is to be able to analyze a site much the way we do as humans by evaluating the site using human metrics and preferences.
Keep in mind that the algorithm change has effected 1%, .7%, and 1% of search making it rather easy for sites that are not top heavy with ads but have wide header space to escape the consequences. Also keep in mind that this is not technically a penalty but a change to you SERP placement. If any site has been hit by this algorithm will not only have to have the site pages re-indexed, but also wait for a refresh of the Panda algorithm making recovery a slow process.
While there is a lot of chatter on the net about "Top Heavy", there is actually little information and testing on this algorithm change and much of the chatter is just SEOs parroting the announcement.
I would pay attention to user experience (UX). Content below the fold does effect UX and can increase your bounce rate. If you experience a higher bounce rate post update, it can be a significant clue. Keep in mind that bounce rate is gauged differently for different content types. A high bounce rate can indicate satisfaction. Some data heavy sites are actually rewarded for this reason. However, an increase in the bounce rate for non-data centric sites can influence SERP placement. This is because bounce rate is a major factor.
Also please understand that any significant change in any site including site formatting can signal to search engines that the site needs to be re-evaluated. If along the way, you see a re-fetch of the entire site. This is an indication that the site is being re-evaluated. This can take a fairly significant amount of time. Maybe 6 months. Google, in these cases, recalcualtes all of the sites metrics and a drop in SERP placement is typically seen as part of the process before properly placing the sites pages in the SERPs.
Pay specific attention to ordinary on-page and on-site SEO work. As well, make sure that your inbound link profile (back links) is as strong as it can be. While keywords are important, they are less of a factor as search engines have become more semantic since 2008. Pay attention to changes in keyword usage and make sure you are not creating a site based upon old SEO keyword centric advice. Make sure you are following SEO techniques from 2014 at least.
Final word: It may have been likely that the site did not perform as well as it should have all these years. You mentioned placement in page 2 of the SERPs. This indicates a moderate to lower performing site. With a history of metrics effecting the site, it will more difficult to move the site up in the SERPs without effecting the metrics over a period of time. If after a reasonable period of time the site does not begin to increase in SERP placement, I would be looking closely to trust signals, authority signals, and improving the sites profile overall in a dramatic and aggressive way. This would include reducing any wide header the site has to make sure that for all screen sizes the content can be seen.