Figures obtained by the party show the percentage of summary criminal cases completed within 26 weeks in justice of the peace courts fell from 74% last October to 64% in April.
In the sheriff courts the figure dropped from 73% to 68% over the same period.
Scottish Court Service (SCS) plans to close 10 sheriff courts and seven justice of the peace courts were accepted by the Scottish Government and backed by MSPs last year.
The first wave of closures took place in November, with courts in Annan, Cumbernauld, Irvine, Motherwell, Dornoch, Kirkcudbright and Rothesay shut. Closures took place in Arbroath, Cupar and Stonehaven in May.
Opponents of the cuts warned the remaining courts would struggle to cope with the increased workload, but the SCS denied the delays were due to closures.
The new figures also show the average time for a summary criminal case to be resolved in a justice of the peace court increased from 152 days in October to 177 days in April.
Over the same period the average sheriff court duration increased from 136 days to 147.
Scottish Conservative justice spokeswoman Margaret Mitchell, who obtained the figures through a series of parliamentary questions, said: "The Scottish Government was adamant there would be no impact on cases once these closures were implemented.
"But clearly this is not the case as evidenced from these figures, and this is a situation which can only get worse as more courts face closure across the country.
"The SNP Government was warned repeatedly that these closures were short sighted, that staff would be overstretched and that cases would be delayed and that's exactly what is now happening in our sheriff and JP courts."
SCS chief executive Eric McQueen insisted the drop in the number of cases being settled within 26 weeks was not due to court closures.
Mr McQueen said: "As we have been very clear, the court closures will result in a re-distribution of only 5% of the overall business to courts with sufficient capacity and no reduction in judicial or court staff."
The increases were instead a reflection of "more proactive policing and prosecution, better evidence gathering and greater confidence amongst victims in reporting certain crimes, including historic crimes", he said.
"To tackle these increases, targeted measures have been agreed across justice organisations, with additional judicial, court and Crown Office and Procurator Fiscal Service staff being deployed. These measures will significantly increase the number of cases settled within 26 weeks and clearly demonstrate that court capacity is available to deal with the increased levels of business."