Unless you include my middle name in the search, a few of the links that come up aren't my profiles, but profiles of people who have the same name. One of them is an Instagram that does not give off the impression I want employers to have of me. Will this harm my job search? And is there anything I can do to make the situation better for myself?
Most recruiters (and intelligent people in general) understand that an internet search for a person's name will likely turn up hits for more than one person. I wouldn't get too worked up about it unless your name is very unique.
You can take preemptive steps when you provide your information to someone (this is best done verbally, and only if you are aware that people actually do confuse you with the others of your name out there):
If you search for my name on the internet, just remember, I'm the one from Sometown. There are a few other John Doe's that usually show up ahead of me in searches.
You should search for your name occasionally to be aware of what might be popping up so you can forewarn someone if there is something you wouldn't want to to be falsely associated with.
Better yet, you can provide some more direct search tips to help them. Provide the URL to your LinkedIn profile, or to your own website or blog, or StackExchange profile, or wherever you have good information about yourself that you are in control of. Of course, this assumes you've got more there than just your name and a picture (especially in LinkedIn profile).
A recruiter will probably start at whatever place you give them, and will probably go from there, but at least you've given them enough background information in your first reference that they should be able to identify you in other contexts.
And of course, take the time to make sure your online presence portrays you in the light you want to be seen in. Lock down the privacy settings on any social media where you might have posted things that you don't want potential employers to come across.