http://elderscrolls.wikia.com/wiki/A_Blade_in_the_Dark

So she walks outside Riverwood city gates and make a stand there,

i try to show the way over the bridge, she remains at the gates.

So how am i suppose to travel with here to Kynesgrove if i want?

keep walking. Sometimes I'll almost be to the junction by the brewery before she catches up.

Walking with her doesn't amount to much excitement.