A fourth Welsh-medium secondary school is likely to open in Cardiff within a decade to cope with growing demand.
Cardiff council leader Huw Thomas also said that one of the current schools - Ysgol Plasmawr - would be expanded within the next two years.
More children will want to start Welsh-medium secondaries within two years than there are places, BBC Wales research found.
Campaigners say the current situation is unsustainable.
BBC Radio Cymru's Manylu programme found that the number of applications for year 7 pupils is already higher than the places available at two of the city's three Welsh-medium secondary schools.
Mr Thomas said it was a story of success and he was delighted that figures were "growing in a wave" and the council would be speaking to the Welsh Government about its duty to meet the demand.
"We've already put one bid in to expand one of the secondary schools in Cardiff by 30 places," he said. "Although that wasn't successful initially we'll be looking at how we can draw in other sources of funding to make sure that provision is there."
Mr Thomas added: "I certainly envisage that over the five-to-10 year period we will certainly be looking at building a fourth secondary school in Cardiff.
"Where exactly and when, I think, is a matter to be worked through over the coming years and in consultation in a way that doesn't de-stabilize the existing three schools."
Manylu has found:
- The three Welsh-medium secondary schools in Cardiff - Glan Taf, Plasmawr and Bro Edern - have 600 spaces for year 7 pupils. Figures uncovered by the programme show these places will all be filled by September 2020.
- There has been an increase of around 16% in the past six years in the number of pupils starting at Welsh-medium primary schools in the city, which is likely to lead to an even greater demand for Welsh medium secondary places.
- Two of Cardiff's Welsh secondary schools currently receive more applications than spaces available.
- Until now there has been surplus capacity at the third school - Ysgol Bro Edern, which opened in 2012. But that school's year 7 intake is also predicated to be full by September 2020, leading to campaigners warning that a significant increase in capacity is needed.
Mabli Jones, co-chair of the Cardiff branch of campaign group Cymdeithas yr Iaith Gymraeg, said there needed to be ambition and leadership from the council.
"The current situation is not sustainable, clearly because if the three secondary schools are going to be full in the next couple of years then we definitely need to look at what the council will do after that," she said.
Lib Dem councillor Rhys Taylor added: "It's worrying that we haven't got to grips with the trends that are showing us already that some schools will have capacity problems in 2019 and again in 2020."
Iwan Pritchard, head teacher at Bro Edern in Llanedeyrn, said his personal opinion was that "we have got to do everything" to ensure that enough places were available for pupils who wanted them.
"Opening a new secondary school is an expensive process and in the current financial situation we need to make sure that the three current schools are financially viable, and that is also as important," he said. "Securing the current three schools is as important as securing provision for all the pupils who are going to receive secondary education within a couple of years."
Ms Jones argues that we should move away from a system where Welsh medium education "is an exception or something that parents have to campaign for".
"The council is not succeeding in meeting the demand that's already there and definitely not succeeding in moving towards a new way of doing things which gives every single child Welsh medium education," she added.
'It certainly felt like a battle for us'
Angharad Naylor lives in Heath but wasn't able to receive a place for her sons, aged seven and four, at their catchment primary school as it has been at capacity for a number of years.
When it was time for her eldest to be allocated a place they went for the next nearest school at the time but the initial application for the youngest at the same school was refused.
"Not being allocated a place at the school where his sibling was was a shock, but a very emotional time as well," she said.
They were allocated a place for the youngest child after an appeal.
"It certainly felt like a battle for us and something we really shouldn't have to battle for," she said. "There were a number of us as parents facing a similar situation where we would have siblings in different schools and how you overcome that is quite stressful and emotional for us as parents.
"There is hope that there will be change. There are new Welsh medium schools within Cardiff opened in recent years which is excellent. I think there is a need to look at Cardiff North in particular and certainly the catchment of Heath area."
The council said it did monitor patterns of take-up in Welsh-medium provision and was committed to see Cardiff "play its full part in contributing to the million of Welsh speakers we want to see by 2050."
There is a commitment that 30% of Welsh children should receive Welsh medium education by 2031, although Cardiff council's own projections show only about 15% of Cardiff secondary pupils will receive their education through the medium of Welsh by that time.
This is similar to the projections for Swansea, Rhondda Cynon Taf and Wrexham.
Manylu can be heard on BBC Radio Cymru at 12:30 GMT on Thursday 14 March and also via the BBC iPlayer.