What's actually plotted in the figure--average or somesuch over each year as a single point? Can't really tell since there are no markers.
I don't see much trend of any sort; linear or not.
If you don't want to consider a couple of years, simply remove those data from the set but keep the actual year--you'll have a hole but that's just the way time is; "it keeps everything from happening at once".
ADDENDUM
That is, remove both the time and the response variables for those years not considered; shorten the overall series by that number of observations, don't just fixup the observations and leave the time values.
