Traffic police slapped a Tk5,000 fine on motorcyclist Abul Kalam Azad for wrong parking in the capital's Mirpur area on Sunday.
Mehedi Hasan faced the same penalty for riding his motorbike without a driving licence on the Bhatara road.
Both of them blamed the Traffic Department of the Dhaka Metropolitan Police (DMP) for "targeting motorbikes and going easy on buses".
"The traffic police now go harsh against motorcycles and private cars. But they do not take actions against buses and other heavy vehicles for violating traffic rules," an angry Azad said.
Mehedi said the police fined him despite having his learner's driving licence because motorcyclists cannot stage protests as truckers do. "They only fear the transport workers who can organise protests," he added.
Traffic police officials denied the allegations.
A deputy commissioner said drivers without licence are not driving buses nowadays.
"We found trivial issues with some buses but ignored those because we have been instructed to take actions only in cases of major violations," he said.
The government had announced to implement the Road Transport Act 2018 from November 1 countrywide, but the traffic police in the capital started enforcing the law from December 1.
The Dhaka Metropolitan Police started imposing fines according to this law on Sunday on a limited scale.
Fines have been imposed for violating traffic signals, driving without licence and on the wrong side, said Mofiz Uddin Ahmed, additional commissioner of traffic.
He said police fined four bikers Tk5,000 each at Bijoy Sarani but did not disclose the total number of cases filed under the Road Transport Act since Sunday.
The traffic department has launched the drive in some 18 areas under the North, West and East traffic divisions, Mofiz added.
Traffic south division's Deputy Commissioner Joydev Chowdhury admitted that they are yet to receive case receipts as per the newly implemented law.
"As soon as we get the papers, we will go into action," he added.
Sher-e-Bangla Nagar traffic zone's Assistant Commissioner Abul Hossain said they are filing cases and imposing fines as per the DMP directives.
"We fined at least eight bikers during the last two days for not having driving licences and wrong-side driving," he added.
Ramna traffic zone's Assistant Commissioner Refatul Islam said they have been instructed to form a committee under each traffic zone comprising an assistant commissioner, two senior traffic inspectors and at least three sergeants to conduct the drives against violators.
"We got all the updated receipt papers on Sunday following the formation of the committee. On Monday, we filed cases against three motorcycle riders in Ramna traffic area," he added.
He also said one of them was charged for driving on the wrong side while the two others for driving without licences.
"We imposed Tk5,000 fine on each driver for not having licence and Tk1,000 for driving on the wrong side," he said.
Sabujbagh traffic zone Assistant Commissioner Ashrafi Tanjina said they ignored minor violations as per the High Court directive.
"The High Court gave orders for not suing drivers for driving heavy vehicles with licence for light vehicles. But we have not spared anyone for major violations," she added.
The much-talked-about Road Transport Act came into effect on November 17.
The Road Transport Bill 2018 was passed by parliament on September 19, 2018, keeping a provision of maximum five years' jail for any death or serious injuries caused by reckless or negligent driving.