Some local areas will face a 20% shortfall in places by 2015, according to analysis of official data from 2012.
Rising birth rates began the squeeze but uncertainty over funds has made it worse, the LGA says.
Unprecedented pressures on new school places
LGA education chairman David Simmonds added that councils were facing "unprecedented pressures" in tackling the shortage of new school places.
The LGA's warning comes as the government opens 93 free schools - 46 with primary places - raising the total to 174 and providing 43,000 new school places.
Free schools and academies are approved directly by the Department for Education, and councils say this can limit their ability to plan strategically, especially as they have no powers to force such schools to expand or close in response to changes in demand.
The LGA wants a ban on any new free schools opening where there is a surplus of places.
Most of the free schools opening this year are in areas under pressure, but not necessarily those under the most pressure.
The government says it has more than doubled funding for new school places and that 70% of free schools were in areas with a basic need - under pressure. But councils say it is the longer term picture that is important because they have to plan ahead for population growth.
Desperate shortage in some areas
The LGA's analysis of local authority data on school-place needs suggests about 1,000 of the 2,277 local school planning districts will be over capacity by 2015-16.
The greatest pressure is focused on about 99 districts, where 20% more pupils are predicted than places.
However, the findings do not take into account some recent steps to increase school places because the analysis is based on official figures from the last academic year.
Overall, two thirds of local authorities predict they will have more pupils than places by the beginning of the 2016 academic year.
Some 40 local councils are predicted to be 10% over capacity, with 15 of those predicting a 20% surplus of pupils over places.