No, linking GPL'ed code makes the product GPL'ed too, unless you're under the System Library exemption, another author-specified exemption or are actually using LGPL'ed not GPL'ed code.
Your code is still under whatever GPL-compatible license you want, but the GPL component means you can only use the product under the GPL, so for example if somebody requested the source code of your distributed product under the GPL, you'd need to provide it, all of it, under the GPL or a compatible license. You cannot just provide the GPL licensed components.
The linked GPL'ed code seems to be client-side/frontend code, so at the very least everything client-side would fall under the GPL (if you want to argue the client(-side) and the server(-side) are different products).
You're thinking of the Affero GPL (AGPL), which triggers the reciprocity clause on modification for network services.
The Lesser GPL (LGPL) limits the reciprocity to the code itself, provided you're only doing dynamic linking or equivalent and not melding the code into yours (i.e. static linking).