It's not possible to eliminate interest groups, in nearly any society, not just democracies, but you can attempt to eliminate or limit their influence over government and/or elections in a few different ways.
First, if we were serious and since this is a major problem in the American system of government (in my opinion) we could move to a complete public financing of all election campaigns. By having no donations at all, and removing the need for candidates to solicit those donations (and ostensibly grant favors and access in return once in office) you free them from all interest group pressures outside of the individual voters.
Second, we could reform the length of terms of office so that they are longer, and candidates and elected officials would not need to constantly run for office, nor need the help of endorsements, fundraising etc. that interest groups can provide.
A final way to limit the influence of these groups is to remove all lobbying from taking place in government buildings, to deny them access to the levers of power, as it were, so that we didn't have such problems with lobbyists for interest groups actually writing the legislation sometimes, and we could get back to the task of simple governance.
All of this is fine and dandy, but removing interest groups altogether short of turning into a dictatorship, is highly unlikely, and some would say they are at least a necessary evil in a representative democracy with a large population and economy such as ours.