Charity Twelves Company, based in Devon, said it had turned away 112 people seeking support in 2015.
A BBC investigation revealed that in Devon and Cornwall police have seen a 55% rise in reports of sexual abuse in the past four years.
Police urged victims to come forward and said they would be protected.
Updates on this story and more from Devon and Cornwall
Between January and November 2015, 1,326 reports of sexual abuse were made to the force, compared with 853 in 2011.
The figures show more than 380 historic allegations - in which the offence was committed before 1 May 2004 - of rape involving a child have been reported to the force since 2011.
Jon Brown, head of development and impact at the NSPCC, said: "Unfortunately there is a significant gap between the number of people who need help and therapy as a result of the abuse, and the availability of services."
He said the rise in reports was a result of historic cases of child sex abuse in the news, which had encouraged more people to come forward.
853 - 2011
888 - 2012
979 - 2013
1,221 - 2014
1,326 - 2015
Mr Brown said the prime minister had highlighted child sexual abuse as a priority along with dealing with terrorism which was "encouraging".
However, Mr Brown said there needed to be a "more joined up approach [between authorities] to provide the provision of help for victims" including "greater treatment and help" and working with schools and families to prevent abuse.
He said local authorities, health services and the police, as well as charities such as the NSPCC, had a duty to protect and care for victims.
Tom Dingwall, chief executive of Twelves Company, which supports sexual and domestic abuse victims in the South West, said it provided services to almost 300 victims in 2015.
"Last year, demand outstripped our capacity to deliver services and we turned away 112 adults from receiving specialist support."
Across Devon and Cornwall there are four Local Safeguarding Children's Boards - which aim to protect those at risk.
Andy Bickley, chair for Plymouth, said investments, including training courses for those who come into contact with children, have been made.
David Taylor, the chair for Devon and Torbay, said authorities were "getting better at identifying abuse".
He said: "It is an extremely complex and challenging area of work, and authorities in the region are learning from best practice in other parts of the country."
Source: Devon and Cornwall Police
Det Ch Insp Neil Ralph said the rise in reports of abuse was a "challenge for the police service, locally and nationally", but the force had made a "huge investment" by creating a dedicated unit and was working with other forces.
"It is a large increase and we've increased our digital and online footprint as these things can be really difficult to report over the phone so it enables them to report abuse [online] much easier.
"People have a high level of trust in the police and the Operation Yewtree impact [means] they have felt more compelled to report abuse to police.
He said a high number of the reports, which were released in response to a freedom of information request by BBC News, related to historical crimes and the force "would do everything it can to investigate them and bring the offenders to justice".
- Local authorities have the statutory responsibility for safeguarding children.
- The safeguarding is overseen by Local Safeguarding Childrenâ€™s Boards.
- These Boards are made up of a number of agencies such as local authorities, police, education & health providers.
- Police will lead on criminal investigations with social care services leading on the safeguarding.
Victims of child sex abuse are being turned away from support services that are being overwhelmed by a large rise in cases.